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CAPS-Maths-Grade-11-1-Exponents-and-surds.html
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<h2> <a href="maths.html"> ⇦ Back to the Maths chapter list</a></h2>
<section class="sv-section">
<h1>Exponents and surds</h1>
<section class="sv-section">
<h2>1. Revision</h2>
</section> <section class="sv-section">
<h2>2. Rational exponents and surds</h2>
<hr><section class="sv-section sv-question sv-table"><br><div class="id1270-2804">ID is: 1270 Seed is: 2804<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form: adding like surds</h3>
</span><div class="problem">
<p>
Simplify the expression shown by collecting like surds where possible.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mroot><mrow><mo>&#x2212;</mo><mtext>16</mtext></mrow><mn>3</mn></mroot><mo>+</mo><mroot><mrow><mo>&#x2212;</mo><mtext>54</mtext></mrow><mn>3</mn></mroot></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 8.317em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.794em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.115em, 1006.79em, 2.696em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mtable" id="MathJax-Span-3"><span style="display: inline-block; position: relative; width: 6.443em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.93em, 1006.44em, 4.511em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 6.443em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1006.44em, 4.511em, -999.997em); top: -3.978em; right: 0em;"><span class="mtd" id="MathJax-Span-4"><span class="mrow" id="MathJax-Span-5"><span class="mroot" id="MathJax-Span-6"><span style="display: inline-block; position: relative; width: 2.638em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.64em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-7"><span class="mo" id="MathJax-Span-8" style="font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-9" style="font-family: STIXGeneral;">16</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.7em, 3.399em, -999.997em); top: -3.978em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.701em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.388em; left: 0.413em;"><span class="mn" id="MathJax-Span-10" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-11" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mroot" id="MathJax-Span-12" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.638em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.64em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-13"><span class="mo" id="MathJax-Span-14" style="font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-15" style="font-family: STIXGeneral;">54</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.7em, 3.399em, -999.997em); top: -3.978em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.701em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.388em; left: 0.413em;"><span class="mn" id="MathJax-Span-16" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.496em; border-left: 0px solid; width: 0px; height: 1.646em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mroot><mrow><mo>β</mo><mtext>16</mtext></mrow><mn>3</mn></mroot><mo>+</mo><mroot><mrow><mo>β</mo><mtext>54</mtext></mrow><mn>3</mn></mroot></mtd></mtr></mtable></math></span></span></div><script type="math/tex; 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\begin{aligned}
\sqrt[3]{-\text{16}} + \sqrt[3]{-\text{54}}
\end{aligned}
</script></div>
<div class="note instructions"><span class="note-header">INSTRUCTIONS:</span>
<ol>
<li>
Do not use a calculator.
</li>
<li>
Write your answer with an exponent. Type <strong>?^(1/2)</strong> for a square root
or <strong>?^(1/3)</strong> for a cube root.
</li>
</ol>
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mrow><mo>&#x2212;</mo><mtext>16</mtext></mrow><mn>3</mn></mroot><mo>+</mo><mroot><mrow><mo>&#x2212;</mo><mtext>54</mtext></mrow><mn>3</mn></mroot><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-17" style="width: 9.019em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.467em, 1007.32em, 3.047em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-18"><span class="mroot" id="MathJax-Span-19"><span style="display: inline-block; position: relative; width: 2.638em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.64em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-20"><span class="mo" id="MathJax-Span-21" style="font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-22" style="font-family: STIXGeneral;">16</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.7em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.701em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.388em; left: 0.413em;"><span class="mn" id="MathJax-Span-23" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-24" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mroot" id="MathJax-Span-25" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.638em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.64em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-26"><span class="mo" id="MathJax-Span-27" style="font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-28" style="font-family: STIXGeneral;">54</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.7em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.701em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.388em; left: 0.413em;"><span class="mn" id="MathJax-Span-29" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-30" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.496em; border-left: 0px solid; width: 0px; height: 1.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mrow><mo>β</mo><mtext>16</mtext></mrow><mn>3</mn></mroot><mo>+</mo><mroot><mrow><mo>β</mo><mtext>54</mtext></mrow><mn>3</mn></mroot><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-2"> \sqrt[3]{-\text{16}} + \sqrt[3]{-\text{54}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="-5 2^(1/3)" size="20">
<div class="response-type">expression</div></div></div></div>
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</div></section><section class="sv-section sv-question sv-table"><br><div class="id1270-9073">ID is: 1270 Seed is: 9073<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form: adding like surds</h3>
</span><div class="problem">
<p>
Simplify the expression shown by collecting like surds where possible.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msqrt><mtext>80</mtext></msqrt><mo>+</mo><msqrt><mtext>125</mtext></msqrt></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-31" style="width: 6.794em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.565em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1005.57em, 2.579em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-32"><span class="mtable" id="MathJax-Span-33"><span style="display: inline-block; position: relative; width: 5.214em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(3.047em, 1005.21em, 4.394em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 5.214em; height: 0px;"><span style="position: absolute; clip: rect(3.047em, 1005.21em, 4.335em, -999.997em); top: -3.92em; right: 0em;"><span class="mtd" id="MathJax-Span-34"><span class="mrow" id="MathJax-Span-35"><span class="msqrt" id="MathJax-Span-36"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-37"><span class="mtext" id="MathJax-Span-38" style="font-family: STIXGeneral;">80</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.92em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.803em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-39" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msqrt" id="MathJax-Span-40" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-41"><span class="mtext" id="MathJax-Span-42" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.92em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.803em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msqrt><mtext>80</mtext></msqrt><mo>+</mo><msqrt><mtext>125</mtext></msqrt></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">
\begin{aligned}
\sqrt{\text{80}} + \sqrt{\text{125}}
\end{aligned}
</script></div>
<div class="note instructions"><span class="note-header">INSTRUCTIONS:</span>
<ol>
<li>
Do not use a calculator.
</li>
<li>
Write your answer with an exponent. Type <strong>?^(1/2)</strong> for a square root
or <strong>?^(1/3)</strong> for a cube root.
</li>
</ol>
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>80</mtext></msqrt><mo>+</mo><msqrt><mtext>125</mtext></msqrt><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-43" style="width: 7.497em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.15em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.525em, 1006.09em, 2.813em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-44"><span class="msqrt" id="MathJax-Span-45"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-46"><span class="mtext" id="MathJax-Span-47" style="font-family: STIXGeneral;">80</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-48" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msqrt" id="MathJax-Span-49" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-50"><span class="mtext" id="MathJax-Span-51" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-52" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.211em; border-left: 0px solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>80</mtext></msqrt><mo>+</mo><msqrt><mtext>125</mtext></msqrt><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-4"> \sqrt{\text{80}} + \sqrt{\text{125}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="9 5^(1/2)" size="20">
<div class="response-type">one-of<div class="response-subtype">type(expression)</div></div></div></div></div>
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</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id1272-3594">ID is: 1272 Seed is: 3594<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Evaluating fractional exponents</h3>
</span><div class="problem">
<p>Evaluate the expression shown. Your answer should be an integer. </p>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em" /><msup><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-53" style="width: 4.16em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.399em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.881em, 1003.4em, 2.345em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-54"><span class="mspace" id="MathJax-Span-55" style="height: 0em; vertical-align: 0em; width: 2.228em; display: inline-block; overflow: hidden;"></span><span class="msubsup" id="MathJax-Span-56"><span style="display: inline-block; position: relative; width: 1.174em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-57" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-58"><span class="mrow" id="MathJax-Span-59"><span class="mfrac" id="MathJax-Span-60"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.18em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-61" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-62" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em"></mspace><msup><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-5"> \qquad 1^{\frac{1}{2}} </script></span>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="1" size="10"> <div class="response-type">numeric</div></div></div></div>
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</div></section><section class="sv-section sv-question sv-table"><br><div class="id1272-4145">ID is: 1272 Seed is: 4145<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Evaluating fractional exponents</h3>
</span><div class="problem">
<p>Evaluate the expression shown. Your answer should be an integer. </p>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em" /><msup><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-63" style="width: 4.16em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.399em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.881em, 1003.4em, 2.345em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-64"><span class="mspace" id="MathJax-Span-65" style="height: 0em; vertical-align: 0em; width: 2.228em; display: inline-block; overflow: hidden;"></span><span class="msubsup" id="MathJax-Span-66"><span style="display: inline-block; position: relative; width: 1.174em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-67" style="font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.447em; left: 0.53em;"><span class="texatom" id="MathJax-Span-68"><span class="mrow" id="MathJax-Span-69"><span class="mfrac" id="MathJax-Span-70"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.18em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-71" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-72" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.575em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em"></mspace><msup><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-6"> \qquad 0^{\frac{1}{3}} </script></span>
</div>
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</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id1274-3680">ID is: 1274 Seed is: 3680<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Evaluating rational exponents</h3>
</span><div class="problem">
<p>
Evaluate the expression shown without using a calculator. Write "non-real" if the answer is not a number.
</p>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em" /><mo stretchy="false">(</mo><mtext>9</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-73" style="width: 4.921em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.043em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1004.04em, 2.872em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-74"><span class="mspace" id="MathJax-Span-75" style="height: 0em; vertical-align: 0em; width: 2.228em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-76" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-77" style="font-family: STIXGeneral;">9</span><span class="msubsup" id="MathJax-Span-78"><span style="display: inline-block; position: relative; width: 0.998em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.3em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mo" id="MathJax-Span-79" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.354em;"><span class="texatom" id="MathJax-Span-80"><span class="mrow" id="MathJax-Span-81"><span class="mfrac" id="MathJax-Span-82"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-83" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-84" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em"></mspace><mo stretchy="false">(</mo><mtext>9</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-7"> \qquad (\text{9} )^{\frac{\text{3} }{\text{2} }} </script></span>
<p>
NOTE: It is easy to type this into a calculator to get the answer. However, it is very important for you to understand what the fraction in the exponent is telling <strong>you</strong>, not your calculator! Also, it is often true that for a question like this you must show your working to get full marks.
</p>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>9</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-85" style="width: 3.457em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.813em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.75em, 2.872em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-86"><span class="mo" id="MathJax-Span-87" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-88" style="font-family: STIXGeneral;">9</span><span class="msubsup" id="MathJax-Span-89"><span style="display: inline-block; position: relative; width: 0.998em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.3em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mo" id="MathJax-Span-90" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.354em;"><span class="texatom" id="MathJax-Span-91"><span class="mrow" id="MathJax-Span-92"><span class="mfrac" id="MathJax-Span-93"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-94" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-95" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-96" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>9</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-8"> (\text{9} )^{\frac{\text{3} }{\text{2} }} = </script></span><input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="27" size="10">
<div class="response-type">numeric</div></div></div></div>
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</div></section><section class="sv-section sv-question sv-table"><br><div class="id1274-935">ID is: 1274 Seed is: 935<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Evaluating rational exponents</h3>
</span><div class="problem">
<p>
Evaluate the expression shown without using a calculator. Write "non-real" if the answer is not a number.
</p>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em" /><mo stretchy="false">(</mo><mtext>64</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>2</mtext><mtext>3</mtext></mfrac></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-97" style="width: 5.565em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.57em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1004.57em, 2.872em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-98"><span class="mspace" id="MathJax-Span-99" style="height: 0em; vertical-align: 0em; width: 2.228em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-100" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-101" style="font-family: STIXGeneral;">64</span><span class="msubsup" id="MathJax-Span-102"><span style="display: inline-block; position: relative; width: 0.998em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.3em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mo" id="MathJax-Span-103" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.447em; left: 0.354em;"><span class="texatom" id="MathJax-Span-104"><span class="mrow" id="MathJax-Span-105"><span class="mfrac" id="MathJax-Span-106"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-107" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-108" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="2em"></mspace><mo stretchy="false">(</mo><mtext>64</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>2</mtext><mtext>3</mtext></mfrac></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-9"> \qquad (\text{64} )^{\frac{\text{2} }{\text{3} }} </script></span>
<p>
NOTE: It is easy to type this into a calculator to get the answer. However, it is very important for you to understand what the fraction in the exponent is telling <strong>you</strong>, not your calculator! Also, it is often true that for a question like this you must show your working to get full marks.
</p>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>64</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>2</mtext><mtext>3</mtext></mfrac></mrow></msup><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-109" style="width: 4.043em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.282em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1003.22em, 2.872em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-110"><span class="mo" id="MathJax-Span-111" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-112" style="font-family: STIXGeneral;">64</span><span class="msubsup" id="MathJax-Span-113"><span style="display: inline-block; position: relative; width: 0.998em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.3em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mo" id="MathJax-Span-114" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.447em; left: 0.354em;"><span class="texatom" id="MathJax-Span-115"><span class="mrow" id="MathJax-Span-116"><span class="mfrac" id="MathJax-Span-117"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-118" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-119" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-120" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>64</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mtext>2</mtext><mtext>3</mtext></mfrac></mrow></msup><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-10"> (\text{64} )^{\frac{\text{2} }{\text{3} }} = </script></span><input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="16" size="10">
<div class="response-type">numeric</div></div></div></div>
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</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id4078-632">ID is: 4078 Seed is: 632<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Rationalising denominators with binomial surds</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2016 Grade 11, P1, Q2.2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>Calculate the value of</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-121" style="width: 8.434em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.912em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.588em, 1006.91em, 3.223em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-122"><span class="mtable" id="MathJax-Span-123"><span style="display: inline-block; position: relative; width: 6.56em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.403em, 1006.56em, 5.038em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 6.56em; height: 0px;"><span style="position: absolute; clip: rect(2.462em, 1006.56em, 5.097em, -999.997em); top: -4.037em; left: 50%; margin-left: -3.276em;"><span class="mtd" id="MathJax-Span-124"><span class="mrow" id="MathJax-Span-125"><span class="mstyle" id="MathJax-Span-126"><span class="mrow" id="MathJax-Span-127"><span class="mfrac" id="MathJax-Span-128"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-129" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-130"><span class="msqrt" id="MathJax-Span-131"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-132"><span class="mtext" id="MathJax-Span-133" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-134" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-135" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-136" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-137" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-138"><span class="mfrac" id="MathJax-Span-139"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.524em;"><span class="mtext" id="MathJax-Span-140" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-141"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-142"><span class="mn" id="MathJax-Span-143" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left: 0px solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-11">
\begin{array}{c}
\dfrac{\text{3}}{\sqrt{\text{3}} + \text{2}} + \dfrac{\text{27}}{\sqrt{27}}
\end{array}
</script></div>
<p><strong>without</strong> using a calculator.</p>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p>The correct <strong>first step</strong> is:</p>
<div class="sv-radio-button-group" name="radio-question1a"><ul class="sv-list"><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aA"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aA" name="question1a" type="radio" value="0" checked=""><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aA">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><msqrt><mn>27</mn></msqrt></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-144" style="width: 17.743em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.523em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1014.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-145"><span class="mtable" id="MathJax-Span-146"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.081em;"><span class="mtd" id="MathJax-Span-147"><span class="mrow" id="MathJax-Span-148"><span class="mstyle" id="MathJax-Span-149"><span class="mrow" id="MathJax-Span-150"><span class="mfrac" id="MathJax-Span-151"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-152" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-153"><span class="msqrt" id="MathJax-Span-154"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-155"><span class="mtext" id="MathJax-Span-156" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-157" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-158" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-159" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-160" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-161"><span class="mfrac" id="MathJax-Span-162"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-163"><span class="msqrt" id="MathJax-Span-164"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-165"><span class="mtext" id="MathJax-Span-166" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-167" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-168" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-169"><span class="msqrt" id="MathJax-Span-170"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-171"><span class="mtext" id="MathJax-Span-172" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-173" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-174" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-175" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-176" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-177"><span class="mfrac" id="MathJax-Span-178"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.524em;"><span class="mtext" id="MathJax-Span-179" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-180"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-181"><span class="mn" id="MathJax-Span-182" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-183" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-184" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-185"><span class="mfrac" id="MathJax-Span-186"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-187"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-188"><span class="mn" id="MathJax-Span-189" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-190"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-191"><span class="mn" id="MathJax-Span-192" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><msqrt><mn>27</mn></msqrt></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-12">
\begin{array}{c}
\dfrac{\text{3}}{\sqrt{\text{3}} + \text{2}} \times \dfrac{\sqrt{\text{3}} - \text{2}}{\sqrt{\text{3}} - \text{2}} + \dfrac{\text{27}}{\sqrt{27}} \times \dfrac{\sqrt{27}}{\sqrt{27}}
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</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aB"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aB" name="question1a" type="radio" value="1"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aB">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><msqrt><mn>27</mn></msqrt></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-193" style="width: 17.743em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.523em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1014.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-194"><span class="mtable" id="MathJax-Span-195"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.081em;"><span class="mtd" id="MathJax-Span-196"><span class="mrow" id="MathJax-Span-197"><span class="mstyle" id="MathJax-Span-198"><span class="mrow" id="MathJax-Span-199"><span class="mfrac" id="MathJax-Span-200"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-201" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-202"><span class="msqrt" id="MathJax-Span-203"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-204"><span class="mtext" id="MathJax-Span-205" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-206" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-207" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-208" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-209" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-210"><span class="mfrac" id="MathJax-Span-211"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-212"><span class="msqrt" id="MathJax-Span-213"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-214"><span class="mtext" id="MathJax-Span-215" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-216" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-217" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-218"><span class="msqrt" id="MathJax-Span-219"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-220"><span class="mtext" id="MathJax-Span-221" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-222" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-223" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-224" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-225" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-226"><span class="mfrac" id="MathJax-Span-227"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.524em;"><span class="mtext" id="MathJax-Span-228" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-229"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-230"><span class="mn" id="MathJax-Span-231" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-232" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-233" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-234"><span class="mfrac" id="MathJax-Span-235"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-236"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-237"><span class="mn" id="MathJax-Span-238" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-239"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-240"><span class="mn" id="MathJax-Span-241" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><msqrt><mn>27</mn></msqrt></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-13">
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\dfrac{\text{3}}{\sqrt{\text{3}} + \text{2}} \times \dfrac{\sqrt{\text{3}} + \text{2}}{\sqrt{\text{3}} + \text{2}} + \dfrac{\text{27}}{\sqrt{27}} \times \dfrac{\sqrt{27}}{\sqrt{27}}
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</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aC"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aC" name="question1a" type="radio" value="2"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aC">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><mn>27</mn></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-242" style="width: 17.743em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.523em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1014.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-243"><span class="mtable" id="MathJax-Span-244"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.081em;"><span class="mtd" id="MathJax-Span-245"><span class="mrow" id="MathJax-Span-246"><span class="mstyle" id="MathJax-Span-247"><span class="mrow" id="MathJax-Span-248"><span class="mfrac" id="MathJax-Span-249"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-250" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-251"><span class="msqrt" id="MathJax-Span-252"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-253"><span class="mtext" id="MathJax-Span-254" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-255" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-256" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-257" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-258" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-259"><span class="mfrac" id="MathJax-Span-260"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-261"><span class="msqrt" id="MathJax-Span-262"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-263"><span class="mtext" id="MathJax-Span-264" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-265" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-266" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-267"><span class="msqrt" id="MathJax-Span-268"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-269"><span class="mtext" id="MathJax-Span-270" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-271" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-272" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-273" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-274" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-275"><span class="mfrac" id="MathJax-Span-276"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.524em;"><span class="mtext" id="MathJax-Span-277" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-278"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-279"><span class="mn" id="MathJax-Span-280" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-281" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-282" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-283"><span class="mfrac" id="MathJax-Span-284"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-285"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-286"><span class="mn" id="MathJax-Span-287" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.276em; left: 50%; margin-left: -0.524em;"><span class="mn" id="MathJax-Span-288" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><mn>27</mn></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-14">
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\dfrac{\text{3}}{\sqrt{\text{3}} + \text{2}} \times \dfrac{\sqrt{\text{3}} + \text{2}}{\sqrt{\text{3}} + \text{2}} + \dfrac{\text{27}}{\sqrt{27}} \times \dfrac{\sqrt{27}}{27}
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</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aD"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aD" name="question1a" type="radio" value="3"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aD">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><mn>27</mn></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-289" style="width: 17.743em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.523em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1014.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-290"><span class="mtable" id="MathJax-Span-291"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 14.172em; height: 0px;"><span style="position: absolute; clip: rect(2.345em, 1014.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.081em;"><span class="mtd" id="MathJax-Span-292"><span class="mrow" id="MathJax-Span-293"><span class="mstyle" id="MathJax-Span-294"><span class="mrow" id="MathJax-Span-295"><span class="mfrac" id="MathJax-Span-296"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-297" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-298"><span class="msqrt" id="MathJax-Span-299"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-300"><span class="mtext" id="MathJax-Span-301" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-302" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-303" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-304" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-305" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-306"><span class="mfrac" id="MathJax-Span-307"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-308"><span class="msqrt" id="MathJax-Span-309"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-310"><span class="mtext" id="MathJax-Span-311" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-312" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-313" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-314"><span class="msqrt" id="MathJax-Span-315"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-316"><span class="mtext" id="MathJax-Span-317" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-318" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-319" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-320" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-321" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-322"><span class="mfrac" id="MathJax-Span-323"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.524em;"><span class="mtext" id="MathJax-Span-324" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-325"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-326"><span class="mn" id="MathJax-Span-327" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-328" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-329" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-330"><span class="mfrac" id="MathJax-Span-331"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-332"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-333"><span class="mn" id="MathJax-Span-334" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.276em; left: 50%; margin-left: -0.524em;"><span class="mn" id="MathJax-Span-335" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>3</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>3</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>27</mn></msqrt><mn>27</mn></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-15">
\begin{array}{c}
\dfrac{\text{3}}{\sqrt{\text{3}} + \text{2}} \times \dfrac{\sqrt{\text{3}} - \text{2}}{\sqrt{\text{3}} - \text{2}} + \dfrac{\text{27}}{\sqrt{27}} \times \dfrac{\sqrt{27}}{27}
\end{array}
</script></div>
</label>
</div></li><li class="sv-list-item" style="display:none;"><div class="sv-radio-button"><input id="hidden" name="question1a" type="radio" value=""></div></li></ul></div>
<p><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-336" style="width: 9.254em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.556em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.998em, 1007.5em, 3.633em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-337"><span class="mstyle" id="MathJax-Span-338"><span class="mrow" id="MathJax-Span-339"><span class="mfrac" id="MathJax-Span-340"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-341" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-342"><span class="msqrt" id="MathJax-Span-343"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-344"><span class="mtext" id="MathJax-Span-345" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-346" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-347" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-348" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-349" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-350"><span class="mfrac" id="MathJax-Span-351"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.524em;"><span class="mtext" id="MathJax-Span-352" style="font-family: STIXGeneral;">27</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -0.875em;"><span class="msqrt" id="MathJax-Span-353"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-354"><span class="mn" id="MathJax-Span-355" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1001.88em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.876em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-356" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.211em; border-left: 0px solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>3</mtext><mrow><msqrt><mtext>3</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>27</mtext><msqrt><mn>27</mn></msqrt></mfrac></mstyle><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-16">
\dfrac{\text{3}}{\sqrt{\text{3}} + \text{2}} + \dfrac{\text{27}}{\sqrt{27}} =
</script></span> <input name="question1b" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="6" size="10"></p>
<div class="response-type">numeric</div></div></div></div>
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</div></section><section class="sv-section sv-question sv-table"><br><div class="id4078-6273">ID is: 4078 Seed is: 6273<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Rationalising denominators with binomial surds</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2016 Grade 11, P1, Q2.2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>Calculate the value of</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x2212;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-357" style="width: 9.019em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.588em, 1007.38em, 3.223em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-358"><span class="mtable" id="MathJax-Span-359"><span style="display: inline-block; position: relative; width: 7.029em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.403em, 1007.03em, 5.038em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.029em; height: 0px;"><span style="position: absolute; clip: rect(2.462em, 1007.03em, 5.097em, -999.997em); top: -4.037em; left: 50%; margin-left: -3.51em;"><span class="mtd" id="MathJax-Span-360"><span class="mrow" id="MathJax-Span-361"><span class="mstyle" id="MathJax-Span-362"><span class="mrow" id="MathJax-Span-363"><span class="mfrac" id="MathJax-Span-364"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.758em;"><span class="mtext" id="MathJax-Span-365" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-366"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-367"><span class="mn" id="MathJax-Span-368" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-369" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mstyle" id="MathJax-Span-370" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-371"><span class="mfrac" id="MathJax-Span-372"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-373" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-374"><span class="msqrt" id="MathJax-Span-375"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-376"><span class="mtext" id="MathJax-Span-377" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-378" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-379" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left: 0px solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>β</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-17">
\begin{array}{c}
\dfrac{\text{125}}{\sqrt{125}} - \dfrac{\text{5}}{\sqrt{\text{5}} + \text{2}}
\end{array}
</script></div>
<p><strong>without</strong> using a calculator.</p>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p>The correct <strong>first step</strong> is:</p>
<div class="sv-radio-button-group" name="radio-question1a"><ul class="sv-list"><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aA"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aA" name="question1a" type="radio" value="0"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aA">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><mn>125</mn></mfrac></mstyle><mo>&#x2212;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-380" style="width: 18.972em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1015.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-381"><span class="mtable" id="MathJax-Span-382"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1015.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px;"><span style="position: absolute; clip: rect(2.286em, 1015.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.55em;"><span class="mtd" id="MathJax-Span-383"><span class="mrow" id="MathJax-Span-384"><span class="mstyle" id="MathJax-Span-385"><span class="mrow" id="MathJax-Span-386"><span class="mfrac" id="MathJax-Span-387"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.758em;"><span class="mtext" id="MathJax-Span-388" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-389"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-390"><span class="mn" id="MathJax-Span-391" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-392" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-393" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-394"><span class="mfrac" id="MathJax-Span-395"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-396"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-397"><span class="mn" id="MathJax-Span-398" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.276em; left: 50%; margin-left: -0.758em;"><span class="mn" id="MathJax-Span-399" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-400" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mstyle" id="MathJax-Span-401" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-402"><span class="mfrac" id="MathJax-Span-403"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-404" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-405"><span class="msqrt" id="MathJax-Span-406"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-407"><span class="mtext" id="MathJax-Span-408" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-409" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-410" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-411" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-412" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-413"><span class="mfrac" id="MathJax-Span-414"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-415"><span class="msqrt" id="MathJax-Span-416"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-417"><span class="mtext" id="MathJax-Span-418" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-419" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-420" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-421"><span class="msqrt" id="MathJax-Span-422"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-423"><span class="mtext" id="MathJax-Span-424" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-425" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-426" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><mn>125</mn></mfrac></mstyle><mo>β</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-18">
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\dfrac{\text{125}}{\sqrt{125}} \times \dfrac{\sqrt{125}}{125} - \dfrac{\text{5}}{\sqrt{\text{5}} + \text{2}} \times \dfrac{\sqrt{\text{5}} - \text{2}}{\sqrt{\text{5}} - \text{2}}
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</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aB"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aB" name="question1a" type="radio" value="1"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aB">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x2212;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-427" style="width: 18.972em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1015.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-428"><span class="mtable" id="MathJax-Span-429"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1015.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px;"><span style="position: absolute; clip: rect(2.286em, 1015.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.55em;"><span class="mtd" id="MathJax-Span-430"><span class="mrow" id="MathJax-Span-431"><span class="mstyle" id="MathJax-Span-432"><span class="mrow" id="MathJax-Span-433"><span class="mfrac" id="MathJax-Span-434"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.758em;"><span class="mtext" id="MathJax-Span-435" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-436"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-437"><span class="mn" id="MathJax-Span-438" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-439" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-440" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-441"><span class="mfrac" id="MathJax-Span-442"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-443"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-444"><span class="mn" id="MathJax-Span-445" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-446"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-447"><span class="mn" id="MathJax-Span-448" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-449" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mstyle" id="MathJax-Span-450" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-451"><span class="mfrac" id="MathJax-Span-452"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-453" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-454"><span class="msqrt" id="MathJax-Span-455"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-456"><span class="mtext" id="MathJax-Span-457" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-458" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-459" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-460" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-461" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-462"><span class="mfrac" id="MathJax-Span-463"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-464"><span class="msqrt" id="MathJax-Span-465"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-466"><span class="mtext" id="MathJax-Span-467" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-468" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-469" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-470"><span class="msqrt" id="MathJax-Span-471"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-472"><span class="mtext" id="MathJax-Span-473" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-474" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-475" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>β</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-19">
\begin{array}{c}
\dfrac{\text{125}}{\sqrt{125}} \times \dfrac{\sqrt{125}}{\sqrt{125}} - \dfrac{\text{5}}{\sqrt{\text{5}} + \text{2}} \times \dfrac{\sqrt{\text{5}} + \text{2}}{\sqrt{\text{5}} - \text{2}}
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</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aC"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aC" name="question1a" type="radio" value="2"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aC">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><mn>125</mn></mfrac></mstyle><mo>&#x2212;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-476" style="width: 18.972em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1015.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-477"><span class="mtable" id="MathJax-Span-478"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1015.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px;"><span style="position: absolute; clip: rect(2.286em, 1015.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.55em;"><span class="mtd" id="MathJax-Span-479"><span class="mrow" id="MathJax-Span-480"><span class="mstyle" id="MathJax-Span-481"><span class="mrow" id="MathJax-Span-482"><span class="mfrac" id="MathJax-Span-483"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.758em;"><span class="mtext" id="MathJax-Span-484" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-485"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-486"><span class="mn" id="MathJax-Span-487" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-488" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-489" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-490"><span class="mfrac" id="MathJax-Span-491"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-492"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-493"><span class="mn" id="MathJax-Span-494" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.276em; left: 50%; margin-left: -0.758em;"><span class="mn" id="MathJax-Span-495" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-496" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mstyle" id="MathJax-Span-497" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-498"><span class="mfrac" id="MathJax-Span-499"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-500" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-501"><span class="msqrt" id="MathJax-Span-502"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-503"><span class="mtext" id="MathJax-Span-504" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-505" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-506" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-507" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-508" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-509"><span class="mfrac" id="MathJax-Span-510"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-511"><span class="msqrt" id="MathJax-Span-512"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-513"><span class="mtext" id="MathJax-Span-514" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-515" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-516" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-517"><span class="msqrt" id="MathJax-Span-518"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-519"><span class="mtext" id="MathJax-Span-520" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-521" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-522" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><mn>125</mn></mfrac></mstyle><mo>β</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-20">
\begin{array}{c}
\dfrac{\text{125}}{\sqrt{125}} \times \dfrac{\sqrt{125}}{125} - \dfrac{\text{5}}{\sqrt{\text{5}} + \text{2}} \times \dfrac{\sqrt{\text{5}} + \text{2}}{\sqrt{\text{5}} - \text{2}}
\end{array}
</script></div>
</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-1aD"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-1aD" name="question1a" type="radio" value="3" checked=""><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-1aD">
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x2212;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>&#x00D7;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>&#x2212;</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-523" style="width: 18.972em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1015.52em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-524"><span class="mtable" id="MathJax-Span-525"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1015.17em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 15.167em; height: 0px;"><span style="position: absolute; clip: rect(2.286em, 1015.17em, 5.097em, -999.997em); top: -3.978em; left: 50%; margin-left: -7.55em;"><span class="mtd" id="MathJax-Span-526"><span class="mrow" id="MathJax-Span-527"><span class="mstyle" id="MathJax-Span-528"><span class="mrow" id="MathJax-Span-529"><span class="mfrac" id="MathJax-Span-530"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.758em;"><span class="mtext" id="MathJax-Span-531" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-532"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-533"><span class="mn" id="MathJax-Span-534" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-535" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-536" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-537"><span class="mfrac" id="MathJax-Span-538"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-539"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-540"><span class="mn" id="MathJax-Span-541" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-542"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-543"><span class="mn" id="MathJax-Span-544" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-545" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mstyle" id="MathJax-Span-546" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-547"><span class="mfrac" id="MathJax-Span-548"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-549" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-550"><span class="msqrt" id="MathJax-Span-551"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-552"><span class="mtext" id="MathJax-Span-553" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-554" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-555" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-556" style="font-family: STIXGeneral; padding-left: 0.237em;">Γ</span><span class="mstyle" id="MathJax-Span-557" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-558"><span class="mfrac" id="MathJax-Span-559"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-560"><span class="msqrt" id="MathJax-Span-561"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-562"><span class="mtext" id="MathJax-Span-563" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-564" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-565" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-566"><span class="msqrt" id="MathJax-Span-567"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-568"><span class="mtext" id="MathJax-Span-569" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-570" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mtext" id="MathJax-Span-571" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msqrt><mn>125</mn></msqrt><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>β</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>Γ</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>5</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow><mrow><msqrt><mtext>5</mtext></msqrt><mo>β</mo><mtext>2</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-21">
\begin{array}{c}
\dfrac{\text{125}}{\sqrt{125}} \times \dfrac{\sqrt{125}}{\sqrt{125}} - \dfrac{\text{5}}{\sqrt{\text{5}} + \text{2}} \times \dfrac{\sqrt{\text{5}} - \text{2}}{\sqrt{\text{5}} - \text{2}}
\end{array}
</script></div>
</label>
</div></li><li class="sv-list-item" style="display:none;"><div class="sv-radio-button"><input id="hidden" name="question1a" type="radio" value=""></div></li></ul></div>
<p><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>&#x2212;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-572" style="width: 9.898em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.083em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.998em, 1008.02em, 3.633em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-573"><span class="mstyle" id="MathJax-Span-574"><span class="mrow" id="MathJax-Span-575"><span class="mfrac" id="MathJax-Span-576"><span style="display: inline-block; position: relative; width: 2.403em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.758em;"><span class="mtext" id="MathJax-Span-577" style="font-family: STIXGeneral;">125</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.109em;"><span class="msqrt" id="MathJax-Span-578"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-579"><span class="mn" id="MathJax-Span-580" style="font-family: STIXGeneral;">125</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.4em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-581" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mstyle" id="MathJax-Span-582" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-583"><span class="mfrac" id="MathJax-Span-584"><span style="display: inline-block; position: relative; width: 3.047em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mtext" id="MathJax-Span-585" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.87em, 4.277em, -999.997em); top: -3.159em; left: 50%; margin-left: -1.461em;"><span class="mrow" id="MathJax-Span-586"><span class="msqrt" id="MathJax-Span-587"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-588"><span class="mtext" id="MathJax-Span-589" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.53em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="font-family: STIXGeneral;">βΎ</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-590" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mtext" id="MathJax-Span-591" style="font-family: STIXGeneral; padding-left: 0.237em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1003.05em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.047em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-592" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.211em; border-left: 0px solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>125</mtext><msqrt><mn>125</mn></msqrt></mfrac></mstyle><mo>β</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mtext>5</mtext><mrow><msqrt><mtext>5</mtext></msqrt><mo>+</mo><mtext>2</mtext></mrow></mfrac></mstyle><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-22">
\dfrac{\text{125}}{\sqrt{125}} - \dfrac{\text{5}}{\sqrt{\text{5}} + \text{2}} =
</script></span> <input name="question1b" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="10" size="10"></p>
<div class="response-type">numeric</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.689274"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id1304-4294">ID is: 1304 Seed is: 4294<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST"><span class="worked_example"><h3 class="sv-heading">Simplifying fractions containing surd expressions</h3>
</span><div class="multi-part"><ol class="multi-part-entries"><li class="multi-part-entry">
<div class="problem">
<p>
Simplify the expression below without using a calculator.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mspace width="2em" /><mstyle displaystyle="true" scriptlevel="0"><mfrac><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>3</mn></mrow></msup><mn>6</mn></mroot><mrow><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mn>6</mn></mroot><mspace width="thickmathspace" /><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msup><mn>6</mn></mroot></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-593" style="width: 8.609em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.029em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.237em, 1007.03em, 3.574em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-594"><span class="mtable" id="MathJax-Span-595"><span style="display: inline-block; position: relative; width: 6.677em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.052em, 1006.68em, 5.389em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 6.677em; height: 0px;"><span style="position: absolute; clip: rect(2.052em, 1006.68em, 5.389em, -999.997em); top: -3.978em; left: 50%; margin-left: -3.334em;"><span class="mtd" id="MathJax-Span-596"><span class="mrow" id="MathJax-Span-597"><span class="mspace" id="MathJax-Span-598" style="height: 0em; vertical-align: 0em; width: 2.228em; display: inline-block; overflow: hidden;"></span><span class="mstyle" id="MathJax-Span-599"><span class="mrow" id="MathJax-Span-600"><span class="mfrac" id="MathJax-Span-601"><span style="display: inline-block; position: relative; width: 4.218em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.755em, 1002.35em, 4.335em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.168em;"><span class="mroot" id="MathJax-Span-602"><span style="display: inline-block; position: relative; width: 2.345em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1001.41em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-603"><span style="display: inline-block; position: relative; width: 1.408em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-604" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-605"><span class="mrow" id="MathJax-Span-606"><span class="mo" id="MathJax-Span-607" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mn" id="MathJax-Span-608" style="font-size: 70.7%; font-family: STIXGeneral;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.47em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.94em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.296em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.647em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.095em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-609" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.755em, 1004.1em, 4.335em, -999.997em); top: -2.924em; left: 50%; margin-left: -2.046em;"><span class="mrow" id="MathJax-Span-610"><span class="mroot" id="MathJax-Span-611"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-612"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-613" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-614"><span class="mrow" id="MathJax-Span-615"><span class="mn" id="MathJax-Span-616" style="font-size: 70.7%; font-family: STIXGeneral;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.94em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.471em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.095em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-617" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-618" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span><span class="mroot" id="MathJax-Span-619"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-620"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-621" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-622"><span class="mrow" id="MathJax-Span-623"><span class="mn" id="MathJax-Span-624" style="font-size: 70.7%; font-family: STIXGeneral;">5</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.94em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.471em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.095em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-625" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1004.22em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.218em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.568em; border-left: 0px solid; width: 0px; height: 3.789em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mspace width="2em"></mspace><mstyle displaystyle="true" scriptlevel="0"><mfrac><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>β</mo><mn>3</mn></mrow></msup><mn>6</mn></mroot><mrow><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mn>6</mn></mroot><mspace width="thickmathspace"></mspace><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msup><mn>6</mn></mroot></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-23">
\begin{array}{c}
\qquad \dfrac{\sqrt[6]{2^{-3}}} {\sqrt[6]{2^{4}} \; \sqrt[6]{2^{5}}}
\end{array}
</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
Give your answer as a fraction.
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>3</mn></mrow></msup><mn>6</mn></mroot><mrow><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mn>6</mn></mroot><mspace width="thickmathspace" /><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msup><mn>6</mn></mroot></mrow></mfrac></mstyle><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-626" style="width: 6.677em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.448em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.588em, 1005.39em, 3.926em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-627"><span class="mstyle" id="MathJax-Span-628"><span class="mrow" id="MathJax-Span-629"><span class="mfrac" id="MathJax-Span-630"><span style="display: inline-block; position: relative; width: 4.218em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.755em, 1002.35em, 4.335em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.168em;"><span class="mroot" id="MathJax-Span-631"><span style="display: inline-block; position: relative; width: 2.345em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1001.41em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-632"><span style="display: inline-block; position: relative; width: 1.408em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-633" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-634"><span class="mrow" id="MathJax-Span-635"><span class="mo" id="MathJax-Span-636" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mn" id="MathJax-Span-637" style="font-size: 70.7%; font-family: STIXGeneral;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.47em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.94em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.296em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.647em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.095em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-638" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.755em, 1004.1em, 4.335em, -999.997em); top: -2.924em; left: 50%; margin-left: -2.046em;"><span class="mrow" id="MathJax-Span-639"><span class="mroot" id="MathJax-Span-640"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-641"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-642" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-643"><span class="mrow" id="MathJax-Span-644"><span class="mn" id="MathJax-Span-645" style="font-size: 70.7%; font-family: STIXGeneral;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.94em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.471em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.095em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-646" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-647" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span><span class="mroot" id="MathJax-Span-648"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-649"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-650" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-651"><span class="mrow" id="MathJax-Span-652"><span class="mn" id="MathJax-Span-653" style="font-size: 70.7%; font-family: STIXGeneral;">5</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.94em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.471em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.095em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-654" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1004.22em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.218em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-655" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.568em; border-left: 0px solid; width: 0px; height: 3.789em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>β</mo><mn>3</mn></mrow></msup><mn>6</mn></mroot><mrow><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mn>6</mn></mroot><mspace width="thickmathspace"></mspace><mroot><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msup><mn>6</mn></mroot></mrow></mfrac></mstyle><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-24"> \dfrac{\sqrt[6]{2^{-3}}} {\sqrt[6]{2^{4}} \; \sqrt[6]{2^{5}}} =</script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="1/4" size="10">
</p>
<div class="response-type">fraction</div></div></div></div>
</li>
<li class="multi-part-entry">
<div class="problem">
<p>
What type of number is the answer?
</p>
</div>
<div class="response"><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<div class="sv-radio-button-group" name="radio-question2a"><ul class="sv-list"><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aA"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aA" name="question2a" type="radio" value="0"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aA">Integer</label></div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aB"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aB" name="question2a" type="radio" value="1" checked=""><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aB">Rational number</label></div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aC"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aC" name="question2a" type="radio" value="2"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aC">Irrational number</label></div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aD"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aD" name="question2a" type="radio" value="3"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aD">None of the above</label>
</div></li><li class="sv-list-item" style="display:none;"><div class="sv-radio-button"><input id="hidden" name="question2a" type="radio" value=""></div></li></ul></div>
</div></div></div>
</li>
</ol></div><input name="response_hash" type="hidden" value="2019-04-16T16:53:32.708621"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form></div></section><section class="sv-section sv-question sv-table"><br><div class="id1304-8180">ID is: 1304 Seed is: 8180<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST"><span class="worked_example"><h3 class="sv-heading">Simplifying fractions containing surd expressions</h3>
</span><div class="multi-part"><ol class="multi-part-entries"><li class="multi-part-entry">
<div class="problem">
<p>
Simplify the expression below without using a calculator.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mspace width="2em" /><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msup><mn>9</mn></mroot><mspace width="thickmathspace" /><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>7</mn></mrow></msup><mn>6</mn></mroot></mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>15</mn></mrow></msup><mn>6</mn></mroot></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-656" style="width: 9.136em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.497em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.296em, 1007.5em, 3.574em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-657"><span class="mtable" id="MathJax-Span-658"><span style="display: inline-block; position: relative; width: 7.146em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.111em, 1007.15em, 5.389em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.146em; height: 0px;"><span style="position: absolute; clip: rect(2.052em, 1007.15em, 5.389em, -999.997em); top: -3.978em; left: 50%; margin-left: -3.569em;"><span class="mtd" id="MathJax-Span-659"><span class="mrow" id="MathJax-Span-660"><span class="mspace" id="MathJax-Span-661" style="height: 0em; vertical-align: 0em; width: 2.228em; display: inline-block; overflow: hidden;"></span><span class="mstyle" id="MathJax-Span-662"><span class="mrow" id="MathJax-Span-663"><span class="mfrac" id="MathJax-Span-664"><span style="display: inline-block; position: relative; width: 4.687em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.755em, 1004.57em, 4.335em, -999.997em); top: -4.681em; left: 50%; margin-left: -2.28em;"><span class="mrow" id="MathJax-Span-665"><span class="mroot" id="MathJax-Span-666"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-667"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-668" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-669"><span class="mrow" id="MathJax-Span-670"><span class="mn" id="MathJax-Span-671" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.94em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.471em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.037em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-672" style="font-size: 50%; font-family: STIXGeneral;">9</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-673" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span><span class="mroot" id="MathJax-Span-674"><span style="display: inline-block; position: relative; width: 2.345em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1001.41em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-675"><span style="display: inline-block; position: relative; width: 1.408em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-676" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-677"><span class="mrow" id="MathJax-Span-678"><span class="mo" id="MathJax-Span-679" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mn" id="MathJax-Span-680" style="font-size: 70.7%; font-family: STIXGeneral;">7</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.47em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.94em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.296em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.647em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.037em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-681" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.755em, 1002.23em, 4.335em, -999.997em); top: -2.924em; left: 50%; margin-left: -1.109em;"><span class="mroot" id="MathJax-Span-682"><span style="display: inline-block; position: relative; width: 2.228em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1001.29em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-683"><span style="display: inline-block; position: relative; width: 1.291em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-684" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-685"><span class="mrow" id="MathJax-Span-686"><span class="mn" id="MathJax-Span-687" style="font-size: 70.7%; font-family: STIXGeneral;">15</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.29em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.291em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.037em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-688" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1004.69em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.687em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.568em; border-left: 0px solid; width: 0px; height: 3.718em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mspace width="2em"></mspace><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msup><mn>9</mn></mroot><mspace width="thickmathspace"></mspace><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>β</mo><mn>7</mn></mrow></msup><mn>6</mn></mroot></mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>15</mn></mrow></msup><mn>6</mn></mroot></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-25">
\begin{array}{c}
\qquad \dfrac{\sqrt[9]{3^{6}} \; \sqrt[6]{3^{-7}}} {\sqrt[6]{3^{15}}}
\end{array}
</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
Give your answer as a fraction.
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msup><mn>9</mn></mroot><mspace width="thickmathspace" /><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>7</mn></mrow></msup><mn>6</mn></mroot></mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>15</mn></mrow></msup><mn>6</mn></mroot></mfrac></mstyle><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-689" style="width: 7.263em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.916em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.588em, 1005.86em, 3.926em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-690"><span class="mstyle" id="MathJax-Span-691"><span class="mrow" id="MathJax-Span-692"><span class="mfrac" id="MathJax-Span-693"><span style="display: inline-block; position: relative; width: 4.687em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.755em, 1004.57em, 4.335em, -999.997em); top: -4.681em; left: 50%; margin-left: -2.28em;"><span class="mrow" id="MathJax-Span-694"><span class="mroot" id="MathJax-Span-695"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-696"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-697" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-698"><span class="mrow" id="MathJax-Span-699"><span class="mn" id="MathJax-Span-700" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1000.94em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.471em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.037em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-701" style="font-size: 50%; font-family: STIXGeneral;">9</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-702" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span><span class="mroot" id="MathJax-Span-703"><span style="display: inline-block; position: relative; width: 2.345em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1001.41em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-704"><span style="display: inline-block; position: relative; width: 1.408em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-705" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-706"><span class="mrow" id="MathJax-Span-707"><span class="mo" id="MathJax-Span-708" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mn" id="MathJax-Span-709" style="font-size: 70.7%; font-family: STIXGeneral;">7</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.47em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.94em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.296em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.647em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.037em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-710" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.755em, 1002.23em, 4.335em, -999.997em); top: -2.924em; left: 50%; margin-left: -1.109em;"><span class="mroot" id="MathJax-Span-711"><span style="display: inline-block; position: relative; width: 2.228em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1001.29em, 4.16em, -999.997em); top: -3.978em; left: 0.94em;"><span class="msubsup" id="MathJax-Span-712"><span style="display: inline-block; position: relative; width: 1.291em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-713" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-714"><span class="mrow" id="MathJax-Span-715"><span class="mn" id="MathJax-Span-716" style="font-size: 70.7%; font-family: STIXGeneral;">15</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.29em, 3.399em, -999.997em); top: -4.213em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.291em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -4.037em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.564em; left: 0.354em;"><span class="mn" id="MathJax-Span-717" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1004.69em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.687em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-718" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.568em; border-left: 0px solid; width: 0px; height: 3.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msup><mn>9</mn></mroot><mspace width="thickmathspace"></mspace><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>β</mo><mn>7</mn></mrow></msup><mn>6</mn></mroot></mrow><mroot><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mn>15</mn></mrow></msup><mn>6</mn></mroot></mfrac></mstyle><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-26"> \dfrac{\sqrt[9]{3^{6}} \; \sqrt[6]{3^{-7}}} {\sqrt[6]{3^{15}}} =</script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="1/27" size="10">
</p>
<div class="response-type">fraction</div></div></div></div>
</li>
<li class="multi-part-entry">
<div class="problem">
<p>
What type of number is the answer?
</p>
</div>
<div class="response"><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<div class="sv-radio-button-group" name="radio-question2a"><ul class="sv-list"><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aA"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aA" name="question2a" type="radio" value="0"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aA">Integer</label></div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aB"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aB" name="question2a" type="radio" value="1" checked=""><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aB">Rational number</label></div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aC"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aC" name="question2a" type="radio" value="2"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aC">Irrational number</label></div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aD"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aD" name="question2a" type="radio" value="3"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aD">None of the above</label>
</div></li><li class="sv-list-item" style="display:none;"><div class="sv-radio-button"><input id="hidden" name="question2a" type="radio" value=""></div></li></ul></div>
</div></div></div>
</li>
</ol></div><input name="response_hash" type="hidden" value="2019-04-16T16:53:32.726738"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form></div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id4309-9243">ID is: 4309 Seed is: 9243<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Multiplying out a difference of two squares with surds</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2015 Grade 11, P1, Q2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>
Simplify for <span class="latex-math-wrapper"><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#x2265;</mo><mn>5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-719" style="width: 2.872em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.345em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em, 1002.29em, 2.813em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-720"><span class="mi" id="MathJax-Span-721" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-722" style="font-family: STIXGeneral; padding-left: 0.296em;">β₯</span><span class="mn" id="MathJax-Span-723" style="font-family: STIXGeneral; padding-left: 0.296em;">5</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.211em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>β₯</mo><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-27">x \geq 5</script></span>:</span>
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>10</mn><mi>x</mi><mo>&#x2212;</mo><mn>25</mn></msqrt></msqrt><mo>&#x22C5;</mo><msqrt><mi>x</mi><mo>&#x2212;</mo><msqrt><mn>10</mn><mi>x</mi><mo>&#x2212;</mo><mn>25</mn></msqrt></msqrt></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-724" style="width: 19.617em; display: inline-block;"><span style="display: inline-block; position: relative; width: 16.045em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.823em, 1016.04em, 3.047em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-725"><span class="mtable" id="MathJax-Span-726"><span style="display: inline-block; position: relative; width: 15.694em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.638em, 1015.69em, 4.862em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 15.694em; height: 0px;"><span style="position: absolute; clip: rect(2.638em, 1015.69em, 4.862em, -999.997em); top: -3.978em; right: 0em;"><span class="mtd" id="MathJax-Span-727"><span class="mrow" id="MathJax-Span-728"><span class="msqrt" id="MathJax-Span-729"><span style="display: inline-block; position: relative; width: 7.439em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1006.38em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-730"><span class="mi" id="MathJax-Span-731" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-732" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msqrt" id="MathJax-Span-733" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 4.687em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1003.63em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-734"><span class="mn" id="MathJax-Span-735" style="font-family: STIXGeneral;">10</span><span class="mi" id="MathJax-Span-736" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-737" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-738" style="font-family: STIXGeneral; padding-left: 0.237em;">25</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1003.75em, 3.399em, -999.997em); top: -3.978em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 3.75em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 3.223em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.764em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.174em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.584em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.994em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.403em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.813em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1006.38em, 3.399em, -999.997em); top: -4.271em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 6.385em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 5.916em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.642em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.111em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.52em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.93em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.34em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.809em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.218em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.628em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.038em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.448em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.569em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-739" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msqrt" id="MathJax-Span-740" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 7.439em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1006.38em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-741"><span class="mi" id="MathJax-Span-742" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-743" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msqrt" id="MathJax-Span-744" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 4.687em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1003.63em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-745"><span class="mn" id="MathJax-Span-746" style="font-family: STIXGeneral;">10</span><span class="mi" id="MathJax-Span-747" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-748" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-749" style="font-family: STIXGeneral; padding-left: 0.237em;">25</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1003.75em, 3.399em, -999.997em); top: -3.978em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 3.75em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 3.223em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.764em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.174em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.584em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.994em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.403em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.813em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1006.38em, 3.399em, -999.997em); top: -4.271em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 6.385em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 5.916em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.642em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.111em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.52em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.93em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.34em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.809em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.218em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.628em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.038em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.448em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.569em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.432em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>10</mn><mi>x</mi><mo>β</mo><mn>25</mn></msqrt></msqrt><mo>β
</mo><msqrt><mi>x</mi><mo>β</mo><msqrt><mn>10</mn><mi>x</mi><mo>β</mo><mn>25</mn></msqrt></msqrt></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-28">
\begin{aligned}
\sqrt{x + \sqrt{10 x - 25}} \cdot \sqrt{x - \sqrt{10 x - 25}}
\end{aligned}
</script></div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>10</mn><mi>x</mi><mo>&#x2212;</mo><mn>25</mn></msqrt></msqrt><mo>&#x22C5;</mo><msqrt><mi>x</mi><mo>&#x2212;</mo><msqrt><mn>10</mn><mi>x</mi><mo>&#x2212;</mo><mn>25</mn></msqrt></msqrt><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-750" style="width: 20.319em; display: inline-block;"><span style="display: inline-block; position: relative; width: 16.631em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.115em, 1016.57em, 3.34em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-751"><span class="msqrt" id="MathJax-Span-752"><span style="display: inline-block; position: relative; width: 7.439em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1006.38em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-753"><span class="mi" id="MathJax-Span-754" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-755" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msqrt" id="MathJax-Span-756" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 4.687em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1003.63em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-757"><span class="mn" id="MathJax-Span-758" style="font-family: STIXGeneral;">10</span><span class="mi" id="MathJax-Span-759" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-760" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-761" style="font-family: STIXGeneral; padding-left: 0.237em;">25</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1003.75em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 3.75em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 3.223em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.764em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.174em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.584em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.994em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.403em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.813em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1006.38em, 3.399em, -999.997em); top: -4.33em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 6.385em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 5.916em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.642em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.111em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.52em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.93em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.34em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.809em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.218em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.628em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.038em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.448em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.627em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-762" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msqrt" id="MathJax-Span-763" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 7.439em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1006.38em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-764"><span class="mi" id="MathJax-Span-765" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-766" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msqrt" id="MathJax-Span-767" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 4.687em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1003.63em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-768"><span class="mn" id="MathJax-Span-769" style="font-family: STIXGeneral;">10</span><span class="mi" id="MathJax-Span-770" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-771" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-772" style="font-family: STIXGeneral; padding-left: 0.237em;">25</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1003.75em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 3.75em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 3.223em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.764em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.174em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.584em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.994em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.403em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.813em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1006.38em, 3.399em, -999.997em); top: -4.33em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 6.385em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 5.916em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.642em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.111em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.52em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.93em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.34em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.809em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.218em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.628em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.038em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 5.448em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.627em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-773" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.854em; border-left: 0px solid; width: 0px; height: 2.432em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>10</mn><mi>x</mi><mo>β</mo><mn>25</mn></msqrt></msqrt><mo>β
</mo><msqrt><mi>x</mi><mo>β</mo><msqrt><mn>10</mn><mi>x</mi><mo>β</mo><mn>25</mn></msqrt></msqrt><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-29">\sqrt{x + \sqrt{10 x - 25}} \cdot \sqrt{x - \sqrt{10 x - 25}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="x - 5" size="20">
<div class="response-type">expression</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.749270"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><section class="sv-section sv-question sv-table"><br><div class="id4309-1708">ID is: 4309 Seed is: 1708<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Multiplying out a difference of two squares with surds</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2015 Grade 11, P1, Q2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>
Simplify for <span class="latex-math-wrapper"><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#x2265;</mo><mn>2</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-774" style="width: 2.872em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.345em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em, 1002.35em, 2.813em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-775"><span class="mi" id="MathJax-Span-776" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-777" style="font-family: STIXGeneral; padding-left: 0.296em;">β₯</span><span class="mn" id="MathJax-Span-778" style="font-family: STIXGeneral; padding-left: 0.296em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.211em; border-left: 0px solid; width: 0px; height: 1.075em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>β₯</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-30">x \geq 2</script></span>:</span>
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>4</mn><mi>x</mi><mo>&#x2212;</mo><mn>4</mn></msqrt></msqrt><mo>&#x22C5;</mo><msqrt><mi>x</mi><mo>&#x2212;</mo><msqrt><mn>4</mn><mi>x</mi><mo>&#x2212;</mo><mn>4</mn></msqrt></msqrt></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-779" style="width: 17.099em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.996em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.823em, 1014em, 3.047em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-780"><span class="mtable" id="MathJax-Span-781"><span style="display: inline-block; position: relative; width: 13.645em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.638em, 1013.64em, 4.862em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 13.645em; height: 0px;"><span style="position: absolute; clip: rect(2.638em, 1013.64em, 4.862em, -999.997em); top: -4.037em; right: 0em;"><span class="mtd" id="MathJax-Span-782"><span class="mrow" id="MathJax-Span-783"><span class="msqrt" id="MathJax-Span-784"><span style="display: inline-block; position: relative; width: 6.443em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1005.39em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-785"><span class="mi" id="MathJax-Span-786" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-787" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msqrt" id="MathJax-Span-788" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1002.7em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-789"><span class="mn" id="MathJax-Span-790" style="font-family: STIXGeneral;">4</span><span class="mi" id="MathJax-Span-791" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-792" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-793" style="font-family: STIXGeneral; padding-left: 0.237em;">4</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.75em, 3.399em, -999.997em); top: -3.978em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.35em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1005.39em, 3.399em, -999.997em); top: -4.271em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 5.389em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 4.921em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.291em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.638em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.106em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.574em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.984em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.453em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.569em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-794" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msqrt" id="MathJax-Span-795" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 6.443em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1005.39em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-796"><span class="mi" id="MathJax-Span-797" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-798" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msqrt" id="MathJax-Span-799" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1002.7em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-800"><span class="mn" id="MathJax-Span-801" style="font-family: STIXGeneral;">4</span><span class="mi" id="MathJax-Span-802" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-803" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-804" style="font-family: STIXGeneral; padding-left: 0.237em;">4</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.75em, 3.399em, -999.997em); top: -3.978em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.35em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1005.39em, 3.399em, -999.997em); top: -4.271em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 5.389em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 4.921em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.291em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.638em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.106em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.574em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.984em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.453em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.569em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.432em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>4</mn><mi>x</mi><mo>β</mo><mn>4</mn></msqrt></msqrt><mo>β
</mo><msqrt><mi>x</mi><mo>β</mo><msqrt><mn>4</mn><mi>x</mi><mo>β</mo><mn>4</mn></msqrt></msqrt></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-31">
\begin{aligned}
\sqrt{x + \sqrt{4 x - 4}} \cdot \sqrt{x - \sqrt{4 x - 4}}
\end{aligned}
</script></div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>4</mn><mi>x</mi><mo>&#x2212;</mo><mn>4</mn></msqrt></msqrt><mo>&#x22C5;</mo><msqrt><mi>x</mi><mo>&#x2212;</mo><msqrt><mn>4</mn><mi>x</mi><mo>&#x2212;</mo><mn>4</mn></msqrt></msqrt><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-805" style="width: 17.86em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.64em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.115em, 1014.58em, 3.34em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-806"><span class="msqrt" id="MathJax-Span-807"><span style="display: inline-block; position: relative; width: 6.443em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1005.39em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-808"><span class="mi" id="MathJax-Span-809" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-810" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msqrt" id="MathJax-Span-811" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1002.7em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-812"><span class="mn" id="MathJax-Span-813" style="font-family: STIXGeneral;">4</span><span class="mi" id="MathJax-Span-814" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-815" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-816" style="font-family: STIXGeneral; padding-left: 0.237em;">4</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.75em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.35em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1005.39em, 3.399em, -999.997em); top: -4.33em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 5.389em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 4.921em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.291em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.638em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.106em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.574em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.984em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.453em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.627em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-817" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msqrt" id="MathJax-Span-818" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 6.443em; height: 0px;"><span style="position: absolute; clip: rect(2.93em, 1005.39em, 4.511em, -999.997em); top: -3.978em; left: 1.057em;"><span class="mrow" id="MathJax-Span-819"><span class="mi" id="MathJax-Span-820" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-821" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msqrt" id="MathJax-Span-822" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1002.7em, 4.277em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-823"><span class="mn" id="MathJax-Span-824" style="font-family: STIXGeneral;">4</span><span class="mi" id="MathJax-Span-825" style="font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-826" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-827" style="font-family: STIXGeneral; padding-left: 0.237em;">4</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.75em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.35em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1005.39em, 3.399em, -999.997em); top: -4.33em; left: 1.057em;"><span style="display: inline-block; position: relative; width: 5.389em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 4.921em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.881em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.291em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.759em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.228em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 2.638em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.106em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.574em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 3.984em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 4.453em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.228em, 1001.12em, 4.453em, -999.997em); top: -3.627em; left: 0em;"><span style="font-family: STIXSizeOneSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-828" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.854em; border-left: 0px solid; width: 0px; height: 2.432em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>x</mi><mo>+</mo><msqrt><mn>4</mn><mi>x</mi><mo>β</mo><mn>4</mn></msqrt></msqrt><mo>β
</mo><msqrt><mi>x</mi><mo>β</mo><msqrt><mn>4</mn><mi>x</mi><mo>β</mo><mn>4</mn></msqrt></msqrt><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-32">\sqrt{x + \sqrt{4 x - 4}} \cdot \sqrt{x - \sqrt{4 x - 4}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="x - 2" size="20">
<div class="response-type">expression</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.771373"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id1273-8059">ID is: 1273 Seed is: 8059<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form</h3>
</span><div class="problem">
<p>
Simplify the surd below without using a calculator.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msqrt><mtext>44</mtext></msqrt></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-829" style="width: 2.579em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.111em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.11em, 2.579em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-830"><span class="mtable" id="MathJax-Span-831"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(3.047em, 1001.76em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.92em; left: 50%; margin-left: -0.875em;"><span class="mtd" id="MathJax-Span-832"><span class="mrow" id="MathJax-Span-833"><span class="msqrt" id="MathJax-Span-834"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-835"><span class="mtext" id="MathJax-Span-836" style="font-family: STIXGeneral;">44</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msqrt><mtext>44</mtext></msqrt></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-33">\begin{array}{c}\sqrt{\text{44}}\end{array}</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
Write your answer with an exponent rather than a root (type <strong>^(1/2)</strong> for a <strong>square root</strong> or <strong>^(1/3)</strong> for a <strong>cube root</strong>).
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>44</mtext></msqrt><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-837" style="width: 3.399em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.525em, 1002.7em, 2.813em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-838"><span class="msqrt" id="MathJax-Span-839"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-840"><span class="mtext" id="MathJax-Span-841" style="font-family: STIXGeneral;">44</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-842" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.211em; border-left: 0px solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>44</mtext></msqrt><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-34">\sqrt{\text{44}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="2 11^(1/2)" size="20">
<div class="response-type">expression</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.789812"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><section class="sv-section sv-question sv-table"><br><div class="id1273-9073">ID is: 1273 Seed is: 9073<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form</h3>
</span><div class="problem">
<p>
Simplify the surd below without using a calculator.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msqrt><mtext>160</mtext></msqrt></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-843" style="width: 3.223em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.638em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.64em, 2.579em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-844"><span class="mtable" id="MathJax-Span-845"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(3.047em, 1002.29em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(2.989em, 1002.29em, 4.277em, -999.997em); top: -3.92em; left: 50%; margin-left: -1.109em;"><span class="mtd" id="MathJax-Span-846"><span class="mrow" id="MathJax-Span-847"><span class="msqrt" id="MathJax-Span-848"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-849"><span class="mtext" id="MathJax-Span-850" style="font-family: STIXGeneral;">160</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msqrt><mtext>160</mtext></msqrt></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-35">\begin{array}{c}\sqrt{\text{160}}\end{array}</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
Write your answer with an exponent rather than a root (type <strong>^(1/2)</strong> for a <strong>square root</strong> or <strong>^(1/3)</strong> for a <strong>cube root</strong>).
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>160</mtext></msqrt><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-851" style="width: 4.043em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.282em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.525em, 1003.22em, 2.813em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-852"><span class="msqrt" id="MathJax-Span-853"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-854"><span class="mtext" id="MathJax-Span-855" style="font-family: STIXGeneral;">160</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-856" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.211em; border-left: 0px solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>160</mtext></msqrt><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-36">\sqrt{\text{160}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="4 10^(1/2)" size="20">
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</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id1269-5858">ID is: 1269 Seed is: 5858<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form</h3>
</span><div class="problem">
<p>
Simplify the surd below. Write your answer with a fractional exponent.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msqrt><mtext>27</mtext></msqrt></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-857" style="width: 2.579em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.111em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.11em, 2.579em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-858"><span class="mtable" id="MathJax-Span-859"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(3.047em, 1001.76em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.92em; left: 50%; margin-left: -0.875em;"><span class="mtd" id="MathJax-Span-860"><span class="mrow" id="MathJax-Span-861"><span class="msqrt" id="MathJax-Span-862"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-863"><span class="mtext" id="MathJax-Span-864" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msqrt><mtext>27</mtext></msqrt></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-37">\begin{array}{c}\sqrt{\text{27}}\end{array}</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
You must put brackets on the exponent for this answer. For example:
<ul>
<li> for the answer <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&#x22C5;</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mspace width="thickmathspace" /></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-865" style="width: 3.399em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.75em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-866"><span class="mn" id="MathJax-Span-867" style="font-family: STIXGeneral;">2</span><span class="mo" id="MathJax-Span-868" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msubsup" id="MathJax-Span-869" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 1.174em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-870" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-871"><span class="mrow" id="MathJax-Span-872"><span class="mfrac" id="MathJax-Span-873"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.18em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-874" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-875" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-876" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>β
</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mspace width="thickmathspace"></mspace></math></span></span><script type="math/tex" id="MathJax-Element-38"> 2 \cdot 5^{\frac{1}{2}} \; </script></span> you should type: 2 * 5^(1/2) </li>
<li> for the answer <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&#x22C5;</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mspace width="thickmathspace" /></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-877" style="width: 3.399em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.75em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-878"><span class="mn" id="MathJax-Span-879" style="font-family: STIXGeneral;">2</span><span class="mo" id="MathJax-Span-880" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msubsup" id="MathJax-Span-881" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 1.174em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-882" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.447em; left: 0.53em;"><span class="texatom" id="MathJax-Span-883"><span class="mrow" id="MathJax-Span-884"><span class="mfrac" id="MathJax-Span-885"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.18em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-886" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-887" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-888" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.575em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>β
</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mspace width="thickmathspace"></mspace></math></span></span><script type="math/tex" id="MathJax-Element-39"> 2 \cdot 5^{\frac{1}{3}} \; </script></span> you should type: 2 * 5^(1/3) </li>
</ul>
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>27</mtext></msqrt><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-889" style="width: 3.399em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.525em, 1002.7em, 2.813em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-890"><span class="msqrt" id="MathJax-Span-891"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-892"><span class="mtext" id="MathJax-Span-893" style="font-family: STIXGeneral;">27</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-894" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.211em; border-left: 0px solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mtext>27</mtext></msqrt><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-40"> \sqrt{\text{27}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="3 3^(1/2)" size="20">
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</div></section><section class="sv-section sv-question sv-table"><br><div class="id1269-4839">ID is: 1269 Seed is: 4839<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form</h3>
</span><div class="problem">
<p>
Simplify the surd below. Write your answer with a fractional exponent.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mroot><mtext>48</mtext><mn>3</mn></mroot></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-895" style="width: 2.579em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.111em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.11em, 2.579em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-896"><span class="mtable" id="MathJax-Span-897"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(3.047em, 1001.76em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(2.989em, 1001.76em, 4.277em, -999.997em); top: -3.92em; left: 50%; margin-left: -0.875em;"><span class="mtd" id="MathJax-Span-898"><span class="mrow" id="MathJax-Span-899"><span class="mroot" id="MathJax-Span-900"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mtext" id="MathJax-Span-901" style="font-family: STIXGeneral;">48</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.447em; left: 0.296em;"><span class="mn" id="MathJax-Span-902" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mroot><mtext>48</mtext><mn>3</mn></mroot></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-41">\begin{array}{c}\sqrt[3]{\text{48}}\end{array}</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
You must put brackets on the exponent for this answer. For example:
<ul>
<li> for the answer <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&#x22C5;</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mspace width="thickmathspace" /></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-903" style="width: 3.399em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.75em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-904"><span class="mn" id="MathJax-Span-905" style="font-family: STIXGeneral;">2</span><span class="mo" id="MathJax-Span-906" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msubsup" id="MathJax-Span-907" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 1.174em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-908" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-909"><span class="mrow" id="MathJax-Span-910"><span class="mfrac" id="MathJax-Span-911"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.18em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-912" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-913" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-914" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>β
</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mspace width="thickmathspace"></mspace></math></span></span><script type="math/tex" id="MathJax-Element-42"> 2 \cdot 5^{\frac{1}{2}} \; </script></span> you should type: 2 * 5^(1/2) </li>
<li> for the answer <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&#x22C5;</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mspace width="thickmathspace" /></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-915" style="width: 3.399em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1002.75em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-916"><span class="mn" id="MathJax-Span-917" style="font-family: STIXGeneral;">2</span><span class="mo" id="MathJax-Span-918" style="font-family: STIXGeneral; padding-left: 0.237em;">β
</span><span class="msubsup" id="MathJax-Span-919" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 1.174em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mn" id="MathJax-Span-920" style="font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.447em; left: 0.53em;"><span class="texatom" id="MathJax-Span-921"><span class="mrow" id="MathJax-Span-922"><span class="mfrac" id="MathJax-Span-923"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.18em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-924" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mn" id="MathJax-Span-925" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mspace" id="MathJax-Span-926" style="height: 0em; vertical-align: 0em; width: 0.296em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.575em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>β
</mo><msup><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mspace width="thickmathspace"></mspace></math></span></span><script type="math/tex" id="MathJax-Element-43"> 2 \cdot 5^{\frac{1}{3}} \; </script></span> you should type: 2 * 5^(1/3) </li>
</ul>
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mtext>48</mtext><mn>3</mn></mroot><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-927" style="width: 3.399em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.755em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.525em, 1002.7em, 2.813em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-928"><span class="mroot" id="MathJax-Span-929"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.94em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mtext" id="MathJax-Span-930" style="font-family: STIXGeneral;">48</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.447em; left: 0.296em;"><span class="mn" id="MathJax-Span-931" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-932" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.211em; border-left: 0px solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mtext>48</mtext><mn>3</mn></mroot><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-44"> \sqrt[3]{\text{48}} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="2 6^(1/3)" size="20">
<div class="response-type">one-of<div class="response-subtype">type(expression.strict)</div></div></div></div></div>
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</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id1271-3903">ID is: 1271 Seed is: 3903<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form: adding like surds</h3>
</span><div class="problem">
<p>
Simplify the expression as much as possible. Note that on a test or exam you must show all of your working to get full marks; you should not use a calculator.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-45-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>&#x2212;</mo><mn>3</mn><mroot><mn>192</mn><mn>3</mn></mroot><mo>+</mo><mn>3</mn><mroot><mn>54</mn><mn>3</mn></mroot><mo>&#x2212;</mo><mn>4</mn><mroot><mn>375</mn><mn>3</mn></mroot></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-933" style="width: 13.586em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.127em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1011.13em, 2.638em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-934"><span class="mtable" id="MathJax-Span-935"><span style="display: inline-block; position: relative; width: 10.776em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(3.047em, 1010.78em, 4.394em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 10.776em; height: 0px;"><span style="position: absolute; clip: rect(2.989em, 1010.78em, 4.277em, -999.997em); top: -3.92em; left: 50%; margin-left: -5.383em;"><span class="mtd" id="MathJax-Span-936"><span class="mrow" id="MathJax-Span-937"><span class="mo" id="MathJax-Span-938" style="font-family: STIXGeneral;">β</span><span class="mn" id="MathJax-Span-939" style="font-family: STIXGeneral;">3</span><span class="mroot" id="MathJax-Span-940"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mn" id="MathJax-Span-941" style="font-family: STIXGeneral;">192</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.447em; left: 0.296em;"><span class="mn" id="MathJax-Span-942" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-943" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mn" id="MathJax-Span-944" style="font-family: STIXGeneral; padding-left: 0.237em;">3</span><span class="mroot" id="MathJax-Span-945"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mn" id="MathJax-Span-946" style="font-family: STIXGeneral;">54</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.447em; left: 0.296em;"><span class="mn" id="MathJax-Span-947" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-948" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-949" style="font-family: STIXGeneral; padding-left: 0.237em;">4</span><span class="mroot" id="MathJax-Span-950"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mn" id="MathJax-Span-951" style="font-family: STIXGeneral;">375</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.447em; left: 0.296em;"><span class="mn" id="MathJax-Span-952" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.425em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>β</mo><mn>3</mn><mroot><mn>192</mn><mn>3</mn></mroot><mo>+</mo><mn>3</mn><mroot><mn>54</mn><mn>3</mn></mroot><mo>β</mo><mn>4</mn><mroot><mn>375</mn><mn>3</mn></mroot></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-45">\begin{array}{c}-3 \sqrt[3]{192} +3 \sqrt[3]{54} -4 \sqrt[3]{375}\end{array}</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
You must type a cube root using an exponent, like this: 5 * 3^(1/3).
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
The answer is: <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="-32 3^(1/3) + 9 2^(1/3)" size="20">.
<div class="response-type">expression</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.880266"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><section class="sv-section sv-question sv-table"><br><div class="id1271-3293">ID is: 1271 Seed is: 3293<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplest surd form: adding like surds</h3>
</span><div class="problem">
<p>
Simplify the expression as much as possible. Note that on a test or exam you must show all of your working to get full marks; you should not use a calculator.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-46-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>4</mn><msqrt><mn>162</mn></msqrt><mo>+</mo><msqrt><mn>200</mn></msqrt><mo>&#x2212;</mo><msqrt><mn>32</mn></msqrt></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-953" style="width: 11.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.429em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1009.43em, 2.579em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-954"><span class="mtable" id="MathJax-Span-955"><span style="display: inline-block; position: relative; width: 9.078em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(3.047em, 1009.08em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 9.078em; height: 0px;"><span style="position: absolute; clip: rect(2.989em, 1009.08em, 4.277em, -999.997em); top: -3.92em; left: 50%; margin-left: -4.564em;"><span class="mtd" id="MathJax-Span-956"><span class="mrow" id="MathJax-Span-957"><span class="mn" id="MathJax-Span-958" style="font-family: STIXGeneral;">4</span><span class="msqrt" id="MathJax-Span-959"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-960"><span class="mn" id="MathJax-Span-961" style="font-family: STIXGeneral;">162</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-962" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msqrt" id="MathJax-Span-963" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001.47em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-964"><span class="mn" id="MathJax-Span-965" style="font-family: STIXGeneral;">200</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.52em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-966" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msqrt" id="MathJax-Span-967" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1001em, 4.16em, -999.997em); top: -3.978em; left: 0.764em;"><span class="mrow" id="MathJax-Span-968"><span class="mn" id="MathJax-Span-969" style="font-family: STIXGeneral;">32</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.06em, 3.399em, -999.997em); top: -3.978em; left: 0.764em;"><span style="display: inline-block; position: relative; width: 1.057em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0.53em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.237em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1000.76em, 4.16em, -999.997em); top: -3.861em; left: 0em;"><span style="font-family: STIXVariants;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>4</mn><msqrt><mn>162</mn></msqrt><mo>+</mo><msqrt><mn>200</mn></msqrt><mo>β</mo><msqrt><mn>32</mn></msqrt></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-46">\begin{array}{c}4 \sqrt{162} + \sqrt{200} - \sqrt{32}\end{array}</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
You can type 'sqrt' for the square root. For example: 5 * sqrt(3).
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
The answer is: <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="42 sqrt(2)" size="20">.
<div class="response-type">expression</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.910324"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id4077-3272">ID is: 4077 Seed is: 3272<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplifying fractions with surds</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2016 Grade 11, P1, Q2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>Simplify fully, <strong>without</strong> using a calculator:</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>20</mtext><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt><mo>&#x2212;</mo><msqrt><mtext>80</mtext><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mrow><msqrt><mtext>125</mtext><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-970" style="width: 9.371em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.673em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.296em, 1007.67em, 3.574em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-971"><span class="mtable" id="MathJax-Span-972"><span style="display: inline-block; position: relative; width: 7.321em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.111em, 1007.32em, 5.389em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.321em; height: 0px;"><span style="position: absolute; clip: rect(2.111em, 1007.32em, 5.389em, -999.997em); top: -3.978em; right: 0em;"><span class="mtd" id="MathJax-Span-973"><span class="mrow" id="MathJax-Span-974"><span class="mstyle" id="MathJax-Span-975"><span class="mrow" id="MathJax-Span-976"><span class="mfrac" id="MathJax-Span-977"><span style="display: inline-block; position: relative; width: 7.087em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.872em, 1006.97em, 4.453em, -999.997em); top: -4.798em; left: 50%; margin-left: -3.451em;"><span class="mrow" id="MathJax-Span-978"><span class="msqrt" id="MathJax-Span-979"><span style="display: inline-block; position: relative; width: 2.872em; height: 0px;"><span style="position: absolute; clip: rect(3.047em, 1001.93em, 4.335em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-980"><span class="mtext" id="MathJax-Span-981" style="font-family: STIXGeneral;">20</span><span class="msubsup" id="MathJax-Span-982"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-983" style="font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.271em; left: 0.53em;"><span class="texatom" id="MathJax-Span-984"><span class="mrow" id="MathJax-Span-985"><span class="mtext" id="MathJax-Span-986" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.99em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.994em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.467em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.115em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-987" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msqrt" id="MathJax-Span-988" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.872em; height: 0px;"><span style="position: absolute; clip: rect(3.047em, 1001.93em, 4.335em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-989"><span class="mtext" id="MathJax-Span-990" style="font-family: STIXGeneral;">80</span><span class="msubsup" id="MathJax-Span-991"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-992" style="font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.271em; left: 0.53em;"><span class="texatom" id="MathJax-Span-993"><span class="mrow" id="MathJax-Span-994"><span class="mtext" id="MathJax-Span-995" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.99em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.994em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.467em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.115em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1003.4em, 4.453em, -999.997em); top: -3.1em; left: 50%; margin-left: -1.695em;"><span class="msqrt" id="MathJax-Span-996"><span style="display: inline-block; position: relative; width: 3.399em; height: 0px;"><span style="position: absolute; clip: rect(3.047em, 1002.46em, 4.335em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-997"><span class="mtext" id="MathJax-Span-998" style="font-family: STIXGeneral;">125</span><span class="msubsup" id="MathJax-Span-999"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-1000" style="font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.271em; left: 0.53em;"><span class="texatom" id="MathJax-Span-1001"><span class="mrow" id="MathJax-Span-1002"><span class="mtext" id="MathJax-Span-1003" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.46em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 2.462em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.994em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.764em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.174em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.584em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1007.09em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 7.087em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.568em; border-left: 0px solid; width: 0px; height: 3.718em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>20</mtext><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt><mo>β</mo><msqrt><mtext>80</mtext><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mrow><msqrt><mtext>125</mtext><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-47">
\begin{aligned}
\dfrac
{\sqrt{\text{20}q^{\text{6}}} - \sqrt{\text{80}q^{\text{6}}}}
{\sqrt{\text{125}q^{\text{6}}}}
\end{aligned}
</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span> Give your answer as a <strong>simplified fraction</strong>.</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p><input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="-2/5" size="10"></p>
<div class="response-type">fraction</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.930550"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><section class="sv-section sv-question sv-table"><br><div class="id4077-4713">ID is: 4077 Seed is: 4713<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplifying fractions with surds</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2016 Grade 11, P1, Q2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>Simplify fully, <strong>without</strong> using a calculator:</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-48-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>27</mtext><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt><mo>&#x2212;</mo><msqrt><mtext>75</mtext><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mrow><msqrt><mtext>12</mtext><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1004" style="width: 9.312em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.614em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.296em, 1007.61em, 3.574em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-1005"><span class="mtable" id="MathJax-Span-1006"><span style="display: inline-block; position: relative; width: 7.263em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.111em, 1007.26em, 5.389em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.263em; height: 0px;"><span style="position: absolute; clip: rect(2.111em, 1007.26em, 5.389em, -999.997em); top: -3.978em; right: 0em;"><span class="mtd" id="MathJax-Span-1007"><span class="mrow" id="MathJax-Span-1008"><span class="mstyle" id="MathJax-Span-1009"><span class="mrow" id="MathJax-Span-1010"><span class="mfrac" id="MathJax-Span-1011"><span style="display: inline-block; position: relative; width: 7.029em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.872em, 1006.91em, 4.453em, -999.997em); top: -4.798em; left: 50%; margin-left: -3.451em;"><span class="mrow" id="MathJax-Span-1012"><span class="msqrt" id="MathJax-Span-1013"><span style="display: inline-block; position: relative; width: 2.872em; height: 0px;"><span style="position: absolute; clip: rect(3.047em, 1001.93em, 4.335em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-1014"><span class="mtext" id="MathJax-Span-1015" style="font-family: STIXGeneral;">27</span><span class="msubsup" id="MathJax-Span-1016"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-1017" style="font-family: STIXGeneral; font-style: italic;">π</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.271em; left: 0.471em;"><span class="texatom" id="MathJax-Span-1018"><span class="mrow" id="MathJax-Span-1019"><span class="mtext" id="MathJax-Span-1020" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.93em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.935em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.467em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1021" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msqrt" id="MathJax-Span-1022" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.872em; height: 0px;"><span style="position: absolute; clip: rect(3.047em, 1001.93em, 4.335em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-1023"><span class="mtext" id="MathJax-Span-1024" style="font-family: STIXGeneral;">75</span><span class="msubsup" id="MathJax-Span-1025"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-1026" style="font-family: STIXGeneral; font-style: italic;">π</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.271em; left: 0.471em;"><span class="texatom" id="MathJax-Span-1027"><span class="mrow" id="MathJax-Span-1028"><span class="mtext" id="MathJax-Span-1029" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.93em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.935em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.467em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.872em, 1002.87em, 4.453em, -999.997em); top: -3.1em; left: 50%; margin-left: -1.461em;"><span class="msqrt" id="MathJax-Span-1030"><span style="display: inline-block; position: relative; width: 2.872em; height: 0px;"><span style="position: absolute; clip: rect(3.047em, 1001.93em, 4.335em, -999.997em); top: -3.978em; left: 0.94em;"><span class="mrow" id="MathJax-Span-1031"><span class="mtext" id="MathJax-Span-1032" style="font-family: STIXGeneral;">12</span><span class="msubsup" id="MathJax-Span-1033"><span style="display: inline-block; position: relative; width: 0.94em; height: 0px;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-1034" style="font-family: STIXGeneral; font-style: italic;">π</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.271em; left: 0.471em;"><span class="texatom" id="MathJax-Span-1035"><span class="mrow" id="MathJax-Span-1036"><span class="mtext" id="MathJax-Span-1037" style="font-size: 70.7%; font-family: STIXGeneral;">6</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1001.93em, 3.399em, -999.997em); top: -4.037em; left: 0.94em;"><span style="display: inline-block; position: relative; width: 1.935em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.467em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.354em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.706em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.057em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.813em, 1000.94em, 4.394em, -999.997em); top: -3.92em; left: 0em;"><span style="font-family: STIXGeneral;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1007.03em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 7.029em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.568em; border-left: 0px solid; width: 0px; height: 3.718em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msqrt><mtext>27</mtext><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt><mo>β</mo><msqrt><mtext>75</mtext><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mrow><msqrt><mtext>12</mtext><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mtext>6</mtext></mrow></msup></msqrt></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-48">
\begin{aligned}
\dfrac
{\sqrt{\text{27}p^{\text{6}}} - \sqrt{\text{75}p^{\text{6}}}}
{\sqrt{\text{12}p^{\text{6}}}}
\end{aligned}
</script></div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p><input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="-1" size="10"></p>
<div class="response-type">fraction</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:32.951626"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id4310-6388">ID is: 4310 Seed is: 6388<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST"><span class="worked_example"><h3 class="sv-heading">For which values will a surd expression be a real number?</h3>
</span><div class="multi-part"><div class="multi-part-header">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2015 Grade 11, P1, Q2.2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>
Given:
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-49-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>C</mi><mo>=</mo><msqrt><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo>&#x2212;</mo><mn>2</mn></mrow><mrow><mi>u</mi><mo>&#x2212;</mo><mn>3</mn></mrow></mfrac></mstyle></msqrt><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>u</mi><mn>2</mn></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1038" style="width: 9.78em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.024em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1008.02em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-1039"><span class="mtable" id="MathJax-Span-1040"><span style="display: inline-block; position: relative; width: 7.673em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1007.67em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.673em; height: 0px;"><span style="position: absolute; clip: rect(2.345em, 1007.67em, 5.155em, -999.997em); top: -3.978em; right: 0em;"><span class="mtd" id="MathJax-Span-1041"><span class="mrow" id="MathJax-Span-1042"><span class="mi" id="MathJax-Span-1043" style="font-family: STIXGeneral; font-style: italic;">πΆ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.061em;"></span></span><span class="mo" id="MathJax-Span-1044" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span><span class="msqrt" id="MathJax-Span-1045" style="padding-left: 0.296em;"><span style="display: inline-block; position: relative; width: 3.633em; height: 0px;"><span style="position: absolute; clip: rect(2.462em, 1002.52em, 4.98em, -999.997em); top: -3.978em; left: 1.115em;"><span class="mrow" id="MathJax-Span-1046"><span class="mstyle" id="MathJax-Span-1047"><span class="mrow" id="MathJax-Span-1048"><span class="mfrac" id="MathJax-Span-1049"><span style="display: inline-block; position: relative; width: 2.228em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1001.17em, 4.277em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.583em;"><span class="mrow" id="MathJax-Span-1050"><span class="mo" id="MathJax-Span-1051" style="font-family: STIXGeneral;">β</span><span class="mn" id="MathJax-Span-1052" style="font-family: STIXGeneral;">2</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1002.05em, 4.277em, -999.997em); top: -3.276em; left: 50%; margin-left: -1.051em;"><span class="mrow" id="MathJax-Span-1053"><span class="mi" id="MathJax-Span-1054" style="font-family: STIXGeneral; font-style: italic;">π’</span><span class="mo" id="MathJax-Span-1055" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-1056" style="font-family: STIXGeneral; padding-left: 0.237em;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.23em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.228em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.52em, 3.399em, -999.997em); top: -4.622em; left: 1.115em;"><span style="display: inline-block; position: relative; width: 2.52em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 1.994em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.584em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(1.759em, 1001.17em, 4.57em, -999.997em); top: -3.393em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1057" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-1058" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-1059"><span class="mfrac" id="MathJax-Span-1060"><span style="display: inline-block; position: relative; width: 0.647em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mi" id="MathJax-Span-1061" style="font-family: STIXGeneral; font-style: italic;">π’</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -3.276em; left: 50%; margin-left: -0.231em;"><span class="mn" id="MathJax-Span-1062" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.65em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.647em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>C</mi><mo>=</mo><msqrt><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo>β</mo><mn>2</mn></mrow><mrow><mi>u</mi><mo>β</mo><mn>3</mn></mrow></mfrac></mstyle></msqrt><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>u</mi><mn>2</mn></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-49">
\begin{aligned}
C = \sqrt{\dfrac{-2}{u - 3}} + \dfrac{u}{2}
\end{aligned}
</script></div>
</div>
<ol class="multi-part-entries"><li class="multi-part-entry">
<div class="problem">
<p>
For what value(s) of <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-50-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1063" style="width: 0.588em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.471em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.876em, 1000.41em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1064"><span class="mi" id="MathJax-Span-1065" style="font-family: STIXGeneral; font-style: italic;">π’</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math></span></span><script type="math/tex" id="MathJax-Element-50">u</script></span> will <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-51-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1066" style="width: 0.881em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.706em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.701em, 1000.71em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1067"><span class="mi" id="MathJax-Span-1068" style="font-family: STIXGeneral; font-style: italic;">πΆ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.061em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-51">C</script></span> be a real number?
</p>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-52-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1069" style="width: 0.588em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.471em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.876em, 1000.41em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1070"><span class="mi" id="MathJax-Span-1071" style="font-family: STIXGeneral; font-style: italic;">π’</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math></span></span><script type="math/tex" id="MathJax-Element-52">u</script></span> <select name="question1a" class="response-query-input" autocomplete="off"><option value="">Select</option><option value="0">β₯</option><option value="1">β€</option><option value="2">></option><option value="3" selected=""><</option><option value="4">=</option><option value="5">β </option><option value="6">Ο΅</option></select> <input name="question1b" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="3" size="10">
<div class="response-type">numeric</div></div></div></div>
</li>
<li class="multi-part-entry">
<div class="problem">
<p>
When <span class="latex-math-wrapper"><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-53-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1072" style="width: 2.755em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.228em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em, 1002.11em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1073"><span class="mi" id="MathJax-Span-1074" style="font-family: STIXGeneral; font-style: italic;">π’</span><span class="mo" id="MathJax-Span-1075" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span><span class="mn" id="MathJax-Span-1076" style="font-family: STIXGeneral; padding-left: 0.296em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-53">u = 1</script></span>,</span> what type of number is <span class="latex-math-wrapper"><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-54-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1077" style="width: 0.881em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.706em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.701em, 1000.71em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1078"><span class="mi" id="MathJax-Span-1079" style="font-family: STIXGeneral; font-style: italic;">πΆ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.061em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-54">C</script></span>?</span>
</p>
</div>
<div class="response"><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<div class="sv-radio-button-group" name="radio-question2a"><ul class="sv-list"><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aA"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aA" name="question2a" type="radio" value="0"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aA">irrational</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aB"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aB" name="question2a" type="radio" value="1"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aB">imaginary</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aC"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aC" name="question2a" type="radio" value="2" checked=""><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aC">rational</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aD"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aD" name="question2a" type="radio" value="3"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aD">non-real</label>
</div></li><li class="sv-list-item" style="display:none;"><div class="sv-radio-button"><input id="hidden" name="question2a" type="radio" value=""></div></li></ul></div>
</div></div></div>
</li>
</ol></div><input name="response_hash" type="hidden" value="2019-04-16T16:53:32.975694"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form></div></section><section class="sv-section sv-question sv-table"><br><div class="id4310-2940">ID is: 4310 Seed is: 2940<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST"><span class="worked_example"><h3 class="sv-heading">For which values will a surd expression be a real number?</h3>
</span><div class="multi-part"><div class="multi-part-header">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2015 Grade 11, P1, Q2.2</em></span> <br>
<a url="/info/maths-formulas" href="https://www.emas.com/info/maths-formulas">Maths formulas</a>
</p>
<p>
Given:
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-55-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>B</mi><mo>=</mo><msqrt><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>4</mn><mrow><mi>v</mi><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac></mstyle></msqrt><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>v</mi><mn>3</mn></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1080" style="width: 9.78em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.024em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.53em, 1008.02em, 3.34em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-1081"><span class="mtable" id="MathJax-Span-1082"><span style="display: inline-block; position: relative; width: 7.673em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.345em, 1007.67em, 5.155em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.673em; height: 0px;"><span style="position: absolute; clip: rect(2.345em, 1007.67em, 5.155em, -999.997em); top: -3.978em; right: 0em;"><span class="mtd" id="MathJax-Span-1083"><span class="mrow" id="MathJax-Span-1084"><span class="mi" id="MathJax-Span-1085" style="font-family: STIXGeneral; font-style: italic;">π΅</span><span class="mo" id="MathJax-Span-1086" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span><span class="msqrt" id="MathJax-Span-1087" style="padding-left: 0.296em;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px;"><span style="position: absolute; clip: rect(2.462em, 1002.52em, 4.98em, -999.997em); top: -3.978em; left: 1.115em;"><span class="mrow" id="MathJax-Span-1088"><span class="mstyle" id="MathJax-Span-1089"><span class="mrow" id="MathJax-Span-1090"><span class="mfrac" id="MathJax-Span-1091"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.106em, 1000.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mn" id="MathJax-Span-1092" style="font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1002.05em, 4.277em, -999.997em); top: -3.276em; left: 50%; margin-left: -1.051em;"><span class="mrow" id="MathJax-Span-1093"><span class="mi" id="MathJax-Span-1094" style="font-family: STIXGeneral; font-style: italic;">π£</span><span class="mo" id="MathJax-Span-1095" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="mn" id="MathJax-Span-1096" style="font-family: STIXGeneral; padding-left: 0.237em;">1</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1002.29em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.286em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.989em, 1002.58em, 3.399em, -999.997em); top: -4.622em; left: 1.115em;"><span style="display: inline-block; position: relative; width: 2.579em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 0em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.978em; left: 2.052em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.413em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 0.823em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.232em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.978em; left: 1.642em;">βΎ<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(1.759em, 1001.17em, 4.57em, -999.997em); top: -3.393em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">β</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1097" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mstyle" id="MathJax-Span-1098" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-1099"><span class="mfrac" id="MathJax-Span-1100"><span style="display: inline-block; position: relative; width: 0.647em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.34em, 1000.47em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -0.231em;"><span class="mi" id="MathJax-Span-1101" style="font-family: STIXGeneral; font-style: italic;">π£</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.276em; left: 50%; margin-left: -0.231em;"><span class="mn" id="MathJax-Span-1102" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.65em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.647em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.282em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>B</mi><mo>=</mo><msqrt><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>4</mn><mrow><mi>v</mi><mo>β</mo><mn>1</mn></mrow></mfrac></mstyle></msqrt><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>v</mi><mn>3</mn></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-55">
\begin{aligned}
B = \sqrt{\dfrac{4}{v - 1}} + \dfrac{v}{3}
\end{aligned}
</script></div>
</div>
<ol class="multi-part-entries"><li class="multi-part-entry">
<div class="problem">
<p>
For what value(s) of <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-56-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1103" style="width: 0.647em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.53em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.876em, 1000.53em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1104"><span class="mi" id="MathJax-Span-1105" style="font-family: STIXGeneral; font-style: italic;">π£</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></span></span><script type="math/tex" id="MathJax-Element-56">v</script></span> will <span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-57-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1106" style="width: 0.881em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.706em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.701em, 1000.71em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1107"><span class="mi" id="MathJax-Span-1108" style="font-family: STIXGeneral; font-style: italic;">π΅</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-57">B</script></span> be a real number?
</p>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-58-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1109" style="width: 0.647em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.53em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.876em, 1000.53em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1110"><span class="mi" id="MathJax-Span-1111" style="font-family: STIXGeneral; font-style: italic;">π£</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></span></span><script type="math/tex" id="MathJax-Element-58">v</script></span> <select name="question1a" class="response-query-input" autocomplete="off"><option value="">Select</option><option value="0">β₯</option><option value="1">β€</option><option value="2" selected="">></option><option value="3"><</option><option value="4">=</option><option value="5">β </option><option value="6">Ο΅</option></select> <input name="question1b" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="1" size="10">
<div class="response-type">numeric</div></div></div></div>
</li>
<li class="multi-part-entry">
<div class="problem">
<p>
When <span class="latex-math-wrapper"><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-59-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>6</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1112" style="width: 2.813em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em, 1002.23em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1113"><span class="mi" id="MathJax-Span-1114" style="font-family: STIXGeneral; font-style: italic;">π£</span><span class="mo" id="MathJax-Span-1115" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span><span class="mn" id="MathJax-Span-1116" style="font-family: STIXGeneral; padding-left: 0.296em;">6</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-59">v = 6</script></span>,</span> what type of number is <span class="latex-math-wrapper"><span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-60-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1117" style="width: 0.881em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.706em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.701em, 1000.71em, 2.696em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1118"><span class="mi" id="MathJax-Span-1119" style="font-family: STIXGeneral; font-style: italic;">π΅</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-60">B</script></span>?</span>
</p>
</div>
<div class="response"><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<div class="sv-radio-button-group" name="radio-question2a"><ul class="sv-list"><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aA"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aA" name="question2a" type="radio" value="0"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aA">rational</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aB"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aB" name="question2a" type="radio" value="1"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aB">non-real</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aC"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aC" name="question2a" type="radio" value="2" checked=""><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aC">irrational</label>
</div></li><li class="sv-list-item"><div class="sv-radio-button" id="radio-node-2aD"><div class="sv-radio-button__input-faux"><input autocomplete="off" class="sv-radio-button__input" id="radio-button-2aD" name="question2a" type="radio" value="3"><span class="sv-radio-button__box"></span></div><label class="sv-radio-button__label" for="radio-button-2aD">imaginary</label>
</div></li><li class="sv-list-item" style="display:none;"><div class="sv-radio-button"><input id="hidden" name="question2a" type="radio" value=""></div></li></ul></div>
</div></div></div>
</li>
</ol></div><input name="response_hash" type="hidden" value="2019-04-16T16:53:33.003032"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form></div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id4079-9839">ID is: 4079 Seed is: 9839<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplifying fractions by factorising: variable exponents</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2016 Grade 11, P1, Q2</em></span> <br>
<a url="/info/maths-formulae" href="https://www.emas.com/info/maths-formulae">Maths formulas</a>
</p>
<p>Simplify fully, <strong>without</strong> using a calculator:</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-61-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msup><mtext>7</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>&#x2212;</mo><mtext>2</mtext></mrow></msup><mo>.</mo><msup><mtext>3</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mtext>+2</mtext></mrow></msup></mrow><mrow><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow></msup><mo>&#x2212;</mo><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>&#x2212;</mo><mtext>1</mtext></mrow></msup><mo>.</mo><mtext>3</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1120" style="width: 8.024em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.56em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.471em, 1006.56em, 3.399em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-1121"><span class="mtable" id="MathJax-Span-1122"><span style="display: inline-block; position: relative; width: 6.209em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.286em, 1006.21em, 5.214em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 6.209em; height: 0px;"><span style="position: absolute; clip: rect(2.228em, 1006.21em, 5.155em, -999.997em); top: -3.978em; left: 50%; margin-left: -3.1em;"><span class="mtd" id="MathJax-Span-1123"><span class="mrow" id="MathJax-Span-1124"><span class="mstyle" id="MathJax-Span-1125"><span class="mrow" id="MathJax-Span-1126"><span class="mfrac" id="MathJax-Span-1127"><span style="display: inline-block; position: relative; width: 5.975em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.93em, 1004.04em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.988em;"><span class="mrow" id="MathJax-Span-1128"><span class="msubsup" id="MathJax-Span-1129"><span style="display: inline-block; position: relative; width: 1.818em; height: 0px;"><span style="position: absolute; clip: rect(3.165em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1130" style="font-family: STIXGeneral;">7</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-1131"><span class="mrow" id="MathJax-Span-1132"><span class="mi" id="MathJax-Span-1133" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-1134" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-1135" style="font-size: 70.7%; font-family: STIXGeneral;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1136" style="font-family: STIXGeneral;">.</span><span class="msubsup" id="MathJax-Span-1137" style="padding-left: 0.179em;"><span style="display: inline-block; position: relative; width: 1.818em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1138" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-1139"><span class="mrow" id="MathJax-Span-1140"><span class="mi" id="MathJax-Span-1141" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mtext" id="MathJax-Span-1142" style="font-size: 70.7%; font-family: STIXGeneral;">+2</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.93em, 1005.8em, 4.277em, -999.997em); top: -3.1em; left: 50%; margin-left: -2.924em;"><span class="mrow" id="MathJax-Span-1143"><span class="msubsup" id="MathJax-Span-1144"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.88em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1145" style="font-family: STIXGeneral;">21</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.998em;"><span class="texatom" id="MathJax-Span-1146"><span class="mrow" id="MathJax-Span-1147"><span class="mi" id="MathJax-Span-1148" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π₯</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1149" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msubsup" id="MathJax-Span-1150" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.286em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.88em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1151" style="font-family: STIXGeneral;">21</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.998em;"><span class="texatom" id="MathJax-Span-1152"><span class="mrow" id="MathJax-Span-1153"><span class="mi" id="MathJax-Span-1154" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π₯</span><span class="mo" id="MathJax-Span-1155" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-1156" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1157" style="font-family: STIXGeneral;">.</span><span class="mtext" id="MathJax-Span-1158" style="font-family: STIXGeneral; padding-left: 0.179em;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1005.98em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.975em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.354em; border-left: 0px solid; width: 0px; height: 3.289em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msup><mtext>7</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>β</mo><mtext>2</mtext></mrow></msup><mo>.</mo><msup><mtext>3</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mtext>+2</mtext></mrow></msup></mrow><mrow><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow></msup><mo>β</mo><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>β</mo><mtext>1</mtext></mrow></msup><mo>.</mo><mtext>3</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-61">
\begin{array}{c}
\dfrac
{
\text{7}^{x -\text{2}}.
\text{3}^{x \text{+2}}}
{
\text{21}^{x} -
\text{21}^{x - \text{1}}.
\text{3}}
\end{array}
</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
Give your answer as a <strong>simplified fraction</strong>.
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p><input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="3/14" size="10"></p>
<div class="response-type">fraction</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:33.026653"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><section class="sv-section sv-question sv-table"><br><div class="id4079-4173">ID is: 4079 Seed is: 4173<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Simplifying fractions by factorising: variable exponents</h3>
</span><div class="problem">
<p>
<span class="colored" style="color:gray"><em>Adapted from DBE Nov 2016 Grade 11, P1, Q2</em></span> <br>
<a url="/info/maths-formulae" href="https://www.emas.com/info/maths-formulae">Maths formulas</a>
</p>
<p>Simplify fully, <strong>without</strong> using a calculator:</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-62-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msup><mtext>3</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi><mtext>+2</mtext></mrow></msup><mo>.</mo><msup><mtext>7</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi><mo>&#x2212;</mo><mtext>2</mtext></mrow></msup></mrow><mrow><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msup><mo>&#x2212;</mo><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi><mo>&#x2212;</mo><mtext>1</mtext></mrow></msup><mo>.</mo><mtext>3</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1159" style="width: 7.907em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.443em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.471em, 1006.44em, 3.399em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-1160"><span class="mtable" id="MathJax-Span-1161"><span style="display: inline-block; position: relative; width: 6.092em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.286em, 1006.09em, 5.214em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 6.092em; height: 0px;"><span style="position: absolute; clip: rect(2.228em, 1006.09em, 5.155em, -999.997em); top: -3.978em; left: 50%; margin-left: -3.042em;"><span class="mtd" id="MathJax-Span-1162"><span class="mrow" id="MathJax-Span-1163"><span class="mstyle" id="MathJax-Span-1164"><span class="mrow" id="MathJax-Span-1165"><span class="mfrac" id="MathJax-Span-1166"><span style="display: inline-block; position: relative; width: 5.858em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(2.93em, 1003.93em, 4.16em, -999.997em); top: -4.681em; left: 50%; margin-left: -1.929em;"><span class="mrow" id="MathJax-Span-1167"><span class="msubsup" id="MathJax-Span-1168"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.41em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1169" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-1170"><span class="mrow" id="MathJax-Span-1171"><span class="mi" id="MathJax-Span-1172" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π</span><span class="mtext" id="MathJax-Span-1173" style="font-size: 70.7%; font-family: STIXGeneral;">+2</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1174" style="font-family: STIXGeneral;">.</span><span class="msubsup" id="MathJax-Span-1175" style="padding-left: 0.179em;"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.165em, 1000.47em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1176" style="font-family: STIXGeneral;">7</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.53em;"><span class="texatom" id="MathJax-Span-1177"><span class="mrow" id="MathJax-Span-1178"><span class="mi" id="MathJax-Span-1179" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π</span><span class="mo" id="MathJax-Span-1180" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-1181" style="font-size: 70.7%; font-family: STIXGeneral;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.93em, 1005.68em, 4.277em, -999.997em); top: -3.1em; left: 50%; margin-left: -2.866em;"><span class="mrow" id="MathJax-Span-1182"><span class="msubsup" id="MathJax-Span-1183"><span style="display: inline-block; position: relative; width: 1.408em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.88em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1184" style="font-family: STIXGeneral;">21</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.998em;"><span class="texatom" id="MathJax-Span-1185"><span class="mrow" id="MathJax-Span-1186"><span class="mi" id="MathJax-Span-1187" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1188" style="font-family: STIXGeneral; padding-left: 0.237em;">β</span><span class="msubsup" id="MathJax-Span-1189" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 2.228em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.88em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mtext" id="MathJax-Span-1190" style="font-family: STIXGeneral;">21</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.998em;"><span class="texatom" id="MathJax-Span-1191"><span class="mrow" id="MathJax-Span-1192"><span class="mi" id="MathJax-Span-1193" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π</span><span class="mo" id="MathJax-Span-1194" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mtext" id="MathJax-Span-1195" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1196" style="font-family: STIXGeneral;">.</span><span class="mtext" id="MathJax-Span-1197" style="font-family: STIXGeneral; padding-left: 0.179em;">3</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1005.86em, 1.232em, -999.997em); top: -1.285em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.858em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.354em; border-left: 0px solid; width: 0px; height: 3.289em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msup><mtext>3</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi><mtext>+2</mtext></mrow></msup><mo>.</mo><msup><mtext>7</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi><mo>β</mo><mtext>2</mtext></mrow></msup></mrow><mrow><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msup><mo>β</mo><msup><mtext>21</mtext><mrow class="MJX-TeXAtom-ORD"><mi>b</mi><mo>β</mo><mtext>1</mtext></mrow></msup><mo>.</mo><mtext>3</mtext></mrow></mfrac></mstyle></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-62">
\begin{array}{c}
\dfrac
{
\text{3}^{b \text{+2}}.
\text{7}^{b -\text{2}}}
{
\text{21}^{b} -
\text{21}^{b - \text{1}}.
\text{3}}
\end{array}
</script></div>
<div class="note instruction"><span class="note-header">INSTRUCTION:</span>
Give your answer as a <strong>simplified fraction</strong>.
</div>
</div>
<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<p><input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="3/14" size="10"></p>
<div class="response-type">fraction</div></div></div></div>
<input name="response_hash" type="hidden" value="2019-04-16T16:53:33.049265"><div class="sv"><div class="sv-form__actions" id="nav-buttons"></div></div></form>
</div></section><hr><section class="sv-section sv-question sv-table"><br><div class="id1275-8844">ID is: 1275 Seed is: 8844<br><form name="questions" action="https://www.emas.com/sandbox/nigeria-chapter/a30c8218-434f-5808-8116-2207847d44a0" method="POST">
<span class="worked_example"><h3 class="sv-heading">Evaluating rational exponents</h3>
</span><div class="problem">
<p>
Evaluate the expression shown without using a calculator. Write "non-real" if the answer is not a number; otherwise give your answer as a fraction.
</p>
<div class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-63-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mspace width="2em" /><mo stretchy="false">(</mo><mtext>49</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup></mtd></mtr></mtable></math>" role="presentation" style="text-align: center; position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1198" style="width: 6.677em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.448em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.057em, 1005.45em, 2.755em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-1199"><span class="mtable" id="MathJax-Span-1200"><span style="display: inline-block; position: relative; width: 5.097em; height: 0px; margin-right: 0.179em; margin-left: 0.179em;"><span style="position: absolute; clip: rect(2.872em, 1005.1em, 4.57em, -999.997em); top: -3.978em; left: 0em;"><span style="display: inline-block; position: relative; width: 5.097em; height: 0px;"><span style="position: absolute; clip: rect(2.696em, 1005.1em, 4.335em, -999.997em); top: -3.744em; left: 50%; margin-left: -2.515em;"><span class="mtd" id="MathJax-Span-1201"><span class="mrow" id="MathJax-Span-1202"><span class="mspace" id="MathJax-Span-1203" style="height: 0em; vertical-align: 0em; width: 2.228em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1204" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1205" style="font-family: STIXGeneral;">49</span><span class="msubsup" id="MathJax-Span-1206"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.3em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mo" id="MathJax-Span-1207" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.354em;"><span class="texatom" id="MathJax-Span-1208"><span class="mrow" id="MathJax-Span-1209"><span class="mo" id="MathJax-Span-1210" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mfrac" id="MathJax-Span-1211"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-1212" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-1213" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.568em; border-left: 0px solid; width: 0px; height: 1.789em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mspace width="2em"></mspace><mo stretchy="false">(</mo><mtext>49</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>β</mo><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-63">\begin{array}{c}\qquad (\text{49} )^{-\frac{\text{3} }{\text{2} }}\end{array}</script></div>
<div class="note"><span class="note-header">NOTE:</span>
It is easy to type this into a calculator to get the answer. However, it is very important for you to understand what the fraction in the exponent is telling <strong>you</strong>, not your calculator! Also, it is often true that for a question like this you must show your working to get full marks.
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<div class="response"><a name="next-response"> </a><div class="response-query"><div class="response-query-body"><span class="emphasis-bold answer">Answer:</span>
<span class="latex-math"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-64-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>49</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup><mo>=</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1214" style="width: 4.687em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.809em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em, 1003.75em, 2.872em, -999.997em); top: -2.515em; left: 0em;"><span class="mrow" id="MathJax-Span-1215"><span class="mo" id="MathJax-Span-1216" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1217" style="font-family: STIXGeneral;">49</span><span class="msubsup" id="MathJax-Span-1218"><span style="display: inline-block; position: relative; width: 1.525em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.3em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mo" id="MathJax-Span-1219" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.388em; left: 0.354em;"><span class="texatom" id="MathJax-Span-1220"><span class="mrow" id="MathJax-Span-1221"><span class="mo" id="MathJax-Span-1222" style="font-size: 70.7%; font-family: STIXGeneral;">β</span><span class="mfrac" id="MathJax-Span-1223"><span style="display: inline-block; position: relative; width: 0.354em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -4.33em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-1224" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.457em, 1000.24em, 4.16em, -999.997em); top: -3.627em; left: 50%; margin-left: -0.114em;"><span class="mtext" id="MathJax-Span-1225" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.823em, 1000.35em, 1.232em, -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.354em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.057em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-1226" style="font-family: STIXGeneral; padding-left: 0.296em;">=</span></span><span style="display: inline-block; width: 0px; height: 2.52em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>49</mtext><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>β</mo><mfrac><mtext>3</mtext><mtext>2</mtext></mfrac></mrow></msup><mo>=</mo></math></span></span><script type="math/tex" id="MathJax-Element-64"> (\text{49} )^{-\frac{\text{3} }{\text{2} }} = </script></span> <input name="question1a" class="response-query-input" type="text" autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" value="1/343" size="10">
<div class="response-type">fraction</div></div></div></div>
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