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<title>Rational Exponents</title>
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<div class="section" id="fwk-redden-ch05_s05" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">5.5</span> Rational Exponents</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch05_s05_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch05_s05_o01" numeration="arabic">
<li>Write expressions with rational exponents in radical form.</li>
<li>Write radical expressions with rational exponents.</li>
<li>Perform operations and simplify expressions with rational exponents.</li>
<li>Perform operations on radicals with different indices.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch05_s05_s01" version="5.0" lang="en">
<h2 class="title editable block">Rational Exponents</h2>
<p class="para block" id="fwk-redden-ch05_s05_s01_p01">So far, exponents have been limited to integers. In this section, we will define what rational (or fractional) exponents mean and how to work with them. All of the rules for exponents developed up to this point apply. In particular, recall the product rule for exponents. Given any rational numbers <em class="emphasis">m</em> and <em class="emphasis">n</em>, we have
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1314" display="block"><mrow><msup><mi>x</mi><mi>m</mi></msup><mo>⋅</mo><msup><mi>x</mi><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow></math>
</span>
For example, if we have an exponent of 1/2, then the product rule for exponents implies the following:
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1315" display="block"><mrow><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><msup><mn>5</mn><mn>1</mn></msup><mo>=</mo><mn>5</mn></mrow></math></span>
Here <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1316" display="inline"><mrow><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span> is one of two equal factors of 5; hence it is a square root of 5, and we can write
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1317" display="block"><mrow><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><msqrt><mn>5</mn></msqrt></mrow></math></span>
Furthermore, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1318" display="inline"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span> is one of three equal factors of 2.
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1319" display="block"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⋅</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⋅</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mn>2</mn><mrow><mn>3</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mn>2</mn><mn>1</mn></msup><mo>=</mo><mn>2</mn></mrow></math></span>
Therefore, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1320" display="inline"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span> is a cube root of 2, and we can write
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1321" display="block"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mroot><mn>2</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>
This is true in general, given any nonzero real number <em class="emphasis">a</em> and integer <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1322" display="inline"><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>,
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1323" display="block"><mrow><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup><mo>=</mo><mroot><mi>a</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></math></span>
In other words, the denominator of a fractional exponent determines the index of an <em class="emphasis">n</em>th root.</p>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p14">Rewrite as a radical.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o01" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1324" display="inline"><mrow><msup><mn>6</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1325" display="inline"><mrow><msup><mn>6</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o02" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1326" display="inline"><mrow><msup><mn>6</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mroot><mn>6</mn><mpadded width="0.4em"><mn>2</mn></mpadded></mroot><mo>=</mo><msqrt><mn>6</mn></msqrt></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1327" display="inline"><mrow><msup><mn>6</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mroot><mn>6</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></li>
</ol>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p15">Rewrite as a radical and then simplify.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o03" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1328" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1329" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o03a" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1330" display="inline"><mrow><mtext> </mtext><msup><mrow><mn>16</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><msqrt><mrow><mn>16</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><msup><mn>4</mn><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mn>4</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1331" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>=</mo><mroot><mrow><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mn>2</mn><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mn>2</mn></mrow></math></span></li>
</ol>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p16">Rewrite as a radical and then simplify.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o05" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1332" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>64</mn><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1333" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>32</mn><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mrow><mn>10</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<p class="para">a.</p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1334" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mtext> </mtext><msup><mrow><mo>(</mo><mrow><mn>64</mn><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><mn>64</mn><msup><mi>x</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><msup><mn>4</mn><mn>3</mn></msup><msup><mi>x</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mtd></mtr><mtr><mtd columnalign="right" style="padding-bottom:10%"></mtd><mtd columnalign="left" style="padding-bottom:10%"><mo>=</mo></mtd><mtd columnalign="left" style="padding-bottom:10%"><mn>4</mn><mi>x</mi></mtd></mtr></mtable></math></span></p>
<p class="para">b.</p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1335" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>32</mn><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mrow><mn>10</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><mtext>−</mtext><mn>32</mn><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mrow><mn>10</mn></mrow></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup><msup><mi>x</mi><mn>5</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s05_s01_p17">Next, consider fractional exponents where the numerator is an integer other than 1. For example, consider the following:</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p18"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1336" display="block"><mrow><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⋅</mo><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⋅</mo><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mn>5</mn><mrow><mn>6</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mn>5</mn><mn>2</mn></msup></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p19">This shows that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1337" display="inline"><mrow><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span> is one of three equal factors of <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1338" display="inline"><mrow><msup><mn>5</mn><mn>2</mn></msup></mrow><mo>.</mo></math></span> In other words, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1339" display="inline"><mrow><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span> is a cube root of <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1340" display="inline"><mrow><msup><mn>5</mn><mn>2</mn></msup></mrow></math></span> and we can write:</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1341" display="block"><mrow><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mroot><mrow><msup><mn>5</mn><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p21">In general, given any nonzero real number <em class="emphasis">a</em> where <em class="emphasis">m</em> and <em class="emphasis">n</em> are positive integers (<span class="inlineequation"><math xml:id="fwk-redden-ch05_m1342" display="inline"><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>),</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p22"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1343" display="block"><mrow><msup><mi>a</mi><mrow><mi>m</mi><mo>/</mo><mi>n</mi></mrow></msup><mo>=</mo><mroot><mrow><msup><mi>a</mi><mi>m</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p23">An expression with a <span class="margin_term"><a class="glossterm">rational exponent</a><span class="glossdef">The fractional exponent <em class="emphasis">m</em>/<em class="emphasis">n</em> that indicates a radical with index <em class="emphasis">n</em> and exponent <em class="emphasis">m</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1344" display="inline"><mrow><msup><mi>a</mi><mrow><mi>m</mi><mo>/</mo><mi>n</mi></mrow></msup><mo>=</mo><mroot><mrow><msup><mi>a</mi><mi>m</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow><mo>.</mo></math></span></span></span> is equivalent to a radical where the denominator is the index and the numerator is the exponent. Any radical expression can be written with a rational exponent, which we call <span class="margin_term"><a class="glossterm">exponential form</a><span class="glossdef">An equivalent expression written using a rational exponent.</span></span>.</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p24"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1345" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>R</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi></mrow></mstyle></mrow></mtd><mtd columnalign="center"><mrow></mrow></mtd><mtd columnalign="center"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="center"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></mtd><mtd columnalign="center"><mrow></mrow></mtd><mtd columnalign="center"><mo>=</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p25">Rewrite as a radical.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o07" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1346" display="inline"><mrow><msup><mn>6</mn><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1347" display="inline"><mrow><msup><mn>3</mn><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o08" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1348" display="inline"><mrow><mtext> </mtext><msup><mn>6</mn><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup><mo>=</mo><mroot><mrow><msup><mn>6</mn><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>36</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1349" display="inline"><mrow><msup><mn>3</mn><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>=</mo><mroot><mrow><msup><mn>3</mn><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>27</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></li>
</ol>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p26">Rewrite as a radical and then simplify.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s05_s01_o09" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1350" display="inline"><mrow><msup><mrow><mn>27</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1351" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>12</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s01_p27">We can often avoid very large integers by working with their prime factorization.</p>
<p class="para">a.</p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1352" display="inline"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mn>27</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mrow><mn>27</mn></mrow><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>3</mn><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>R</mi><mi>e</mi><mi>p</mi><mi>l</mi><mi>a</mi><mi>c</mi><mi>e</mi><mtext> </mtext><mn>27</mn><mtext> </mtext><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mtext> </mtext><msup><mn>3</mn><mn>3</mn></msup><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>3</mn><mn>6</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>3</mn><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right" style="padding-bottom:10%"><mrow></mrow></mtd><mtd columnalign="left" style="padding-bottom:10%"><mo>=</mo></mtd><mtd columnalign="left" style="padding-bottom:10%"><mn>9</mn></mtd><mtd columnalign="left" style="padding-bottom:10%"><mrow></mrow></mtd><mtd columnalign="left" style="padding-bottom:10%"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para">b.</p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1353" display="inline"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>12</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>12</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>R</mi><mi>e</mi><mi>p</mi><mi>l</mi><mi>a</mi><mi>c</mi><mi>e</mi><mtext> </mtext><mn>12</mn><mtext> </mtext><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mtext> </mtext><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><mn>3.</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>2</mn><mrow><mn>10</mn></mrow></msup><mo>⋅</mo><msup><mn>3</mn><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>2</mn><mn>9</mn></msup><mo>⋅</mo><mn>2</mn><mo>⋅</mo><msup><mn>3</mn><mn>3</mn></msup><mo>⋅</mo><msup><mn>3</mn><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mn>3</mn></msup><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mroot><mrow><mn>2</mn><mo>⋅</mo><msup><mn>3</mn><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>24</mn><mtext> </mtext><mroot><mrow><mn>18</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s05_s01_p28">Given a radical expression, we might want to find the equivalent in exponential form. Assume all variables are positive.</p>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n06">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p29">Rewrite using rational exponents: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1354" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s01_p30">Here the index is 5 and the power is 3. We can write</p>
<p class="para" id="fwk-redden-ch05_s05_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1355" display="block"><mrow><mroot><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s05_s01_p32">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1356" display="inline"><mrow><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n07">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p33">Rewrite using rational exponents: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1357" display="inline"><mrow><mroot><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s01_p34">Here the index is 6 and the power is 3. We can write</p>
<p class="para" id="fwk-redden-ch05_s05_s01_p35"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1358" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mroot><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>y</mi><mrow><mn>3</mn><mo>/</mo><mn>6</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s05_s01_p36">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1359" display="inline"><mrow><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s05_s01_p37">It is important to note that the following are equivalent.</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p38"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1360" display="block"><mrow><msup><mi>a</mi><mrow><mi>m</mi><mo>/</mo><mi>n</mi></mrow></msup><mo>=</mo><mroot><mrow><msup><mi>a</mi><mi>m</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mroot><mi>a</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mi>m</mi></msup></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch05_s05_s01_p39">In other words, it does not matter if we apply the power first or the root first. For example, we can apply the power before the <em class="emphasis">n</em>th root:</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p40"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1361" display="block"><mrow><msup><mrow><mn>27</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mroot><mrow><msup><mrow><mn>27</mn></mrow><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>3</mn><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mn>3</mn><mn>6</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><msup><mn>3</mn><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s05_s01_p41">Or we can apply the <em class="emphasis">n</em>th root before the power:</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p42"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1362" display="block"><mrow><msup><mrow><mn>27</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mroot><mrow><mn>27</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mroot><mrow><msup><mn>3</mn><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s05_s01_p43">The results are the same.</p>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n08">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p44">Rewrite as a radical and then simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1363" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s01_p45">Here the index is 3 and the power is 2. We can write</p>
<p class="para" id="fwk-redden-ch05_s05_s01_p46"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1364" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mroot><mrow><mtext>−</mtext><mn>8</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>4</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s05_s01_p47">Answer: 4</p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s01_n08a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s05_s01_p48"><strong class="emphasis bold">Try this!</strong> Rewrite as a radical and then simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1365" display="inline"><mrow><msup><mrow><mn>100</mn></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s05_s01_p49">Answer: 1,000</p>
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</div>
</div>
<p class="para block" id="fwk-redden-ch05_s05_s01_p51">Some calculators have a caret button <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1366" display="inline"><mrow><mtable columnspacing="0.1em" frame="solid"><mrow><mtext> </mtext><mo>^</mo><mtext> </mtext></mrow></mtable></mrow></math></span> which is used for entering exponents. If so, we can calculate approximations for radicals using it and rational exponents. For example, to calculate <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1367" display="inline"><mrow><msqrt><mn>2</mn></msqrt><mo>=</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>2</mn><mo>^</mo><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≈</mo><mn>1.414</mn></mrow></math></span>, we make use of the parenthesis buttons and type</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p52"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1368" display="block"><mrow><mn>2</mn><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mo>^</mo></mtable><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mtext> </mtext></mrow></mtable><mtext> </mtext><mtext> </mtext><mn>1</mn><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mo>÷</mo></mtable><mtext> </mtext><mtext> </mtext><mn>2</mn><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mrow><mtext> </mtext><mo stretchy="false">)</mo><mtext> </mtext></mrow></mtable><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mo>=</mo></mtable></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p53">To calculate <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1369" display="inline"><mrow><mroot><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><msup><mn>2</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>2</mn><mo>^</mo><mo stretchy="false">(</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo stretchy="false">)</mo><mo>≈</mo><mn>1.587</mn></mrow></math></span>, we would type</p>
<p class="para block" id="fwk-redden-ch05_s05_s01_p54"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1370" display="block"><mrow><mn>2</mn><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mo>^</mo></mtable><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mtext> </mtext></mrow></mtable><mtext> </mtext><mtext> </mtext><mn>2</mn><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mo>÷</mo></mtable><mtext> </mtext><mtext> </mtext><mn>3</mn><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mrow><mtext> </mtext><mo stretchy="false">)</mo><mtext> </mtext></mrow></mtable><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mo>=</mo></mtable></mrow></math>
</span></p>
</div>
<div class="section" id="fwk-redden-ch05_s05_s02" version="5.0" lang="en">
<h2 class="title editable block">Operations Using the Rules of Exponents</h2>
<p class="para editable block" id="fwk-redden-ch05_s05_s02_p01">In this section, we review all of the rules of exponents, which extend to include rational exponents. If given any rational numbers <em class="emphasis">m</em> and <em class="emphasis">n</em>, then we have</p>
<p class="para block" id="fwk-redden-ch05_s05_s02_p02">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para"><em class="emphasis bolditalic">Product rule for exponents:</em></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1371" display="inline"><mrow><msup><mi>x</mi><mi>m</mi></msup><mo>⋅</mo><msup><mi>x</mi><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><em class="emphasis bolditalic">Quotient rule for exponents:</em></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1372" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mi>m</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>−</mo><mi>n</mi></mrow></msup><mo>,</mo><mtext> </mtext><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><em class="emphasis bolditalic">Power rule for exponents:</em></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1373" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>m</mi></msup></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>⋅</mo><mi>n</mi></mrow></msup></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><em class="emphasis bolditalic">Power rule for a product:</em></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1374" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>y</mi><mi>n</mi></msup></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><em class="emphasis bolditalic">Power rule for a quotient:</em></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1375" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mi>y</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow><mrow><msup><mi>y</mi><mi>n</mi></msup></mrow></mfrac><mtext> </mtext><mo>,</mo><mtext> </mtext><mi>y</mi><mo>≠</mo><mn>0</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><em class="emphasis bolditalic">Negative exponents:</em></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1376" display="inline"><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><em class="emphasis bolditalic">Zero exponent:</em></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1377" display="inline"><mrow><msup><mi>x</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><mtext> </mtext><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch05_s05_s02_p03">These rules allow us to perform operations with rational exponents.</p>
<div class="callout block" id="fwk-redden-ch05_s05_s02_n01">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch05_s05_s02_p04">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1378" display="inline"><mrow><msup><mn>7</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⋅</mo><msup><mn>7</mn><mrow><mn>4</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s02_p05"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1379" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mn>7</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⋅</mo><msup><mn>7</mn><mrow><mn>4</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>7</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn><mo>+</mo><mn>4</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>d</mi><mi>u</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><msup><mi>x</mi><mi>m</mi></msup><mo>⋅</mo><msup><mi>x</mi><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mtext>.</mtext></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>7</mn><mrow><mn>3</mn><mo>/</mo><mn>9</mn><mo>+</mo><mn>4</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>7</mn><mrow><mn>7</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s05_s02_p06">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1380" display="inline"><mrow><msup><mn>7</mn><mrow><mn>7</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s02_n02">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch05_s05_s02_p07">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1381" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1382" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn><mo>−</mo><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mfrac><mrow><msup><mi>x</mi><mi>m</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>−</mo><mi>n</mi></mrow></msup><mtext> </mtext><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mrow><mn>9</mn><mo>/</mo><mn>6</mn><mo>−</mo><mn>4</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s05_s02_p09">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1383" display="inline"><mrow><msup><mi>x</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s02_n03">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch05_s05_s02_p10">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1384" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s02_p11"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1385" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>y</mi><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></msup></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>m</mi></msup></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>⋅</mo><mi>n</mi></mrow></msup><mtext> </mtext><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>y</mi><mrow><mn>6</mn><mo>/</mo><mn>12</mn></mrow></msup></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>M</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>r</mi><mi>e</mi><mi>d</mi><mi>u</mi><mi>c</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s05_s02_p12">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1386" display="inline"><mrow><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s02_n04">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch05_s05_s02_p13">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1387" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>81</mn><msup><mi>a</mi><mn>8</mn></msup><msup><mi>b</mi><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s02_p14"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1388" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>81</mn><msup><mi>a</mi><mn>8</mn></msup><msup><mi>b</mi><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>3</mn><mn>4</mn></msup><msup><mi>a</mi><mn>8</mn></msup><msup><mi>b</mi><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>R</mi><mi>e</mi><mi>w</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mn>81</mn><mtext> </mtext><mi>a</mi><mi>s</mi><mtext> </mtext><msup><mn>3</mn><mn>4</mn></msup><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>3</mn><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>8</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>b</mi><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>d</mi><mi>u</mi><mi>c</mi><mi>t</mi><mtext> </mtext><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>y</mi><mi>n</mi></msup><mtext> </mtext><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>3</mn><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></msup><msup><mi>a</mi><mrow><mn>8</mn><mrow><mo>(</mo><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></msup><msup><mi>b</mi><mrow><mn>12</mn><mrow><mo>(</mo><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mi>e</mi><mi>a</mi><mi>c</mi><mi>h</mi><mtext> </mtext><mi>f</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>3</mn><mn>3</mn></msup><msup><mi>a</mi><mn>6</mn></msup><msup><mi>b</mi><mn>9</mn></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>27</mn><msup><mi>a</mi><mn>6</mn></msup><msup><mi>b</mi><mn>9</mn></msup></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s05_s02_p15">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1389" display="inline"><mrow><mn>27</mn><msup><mi>a</mi><mn>6</mn></msup><msup><mi>b</mi><mn>9</mn></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s02_n05">
<h3 class="title">Example 13</h3>
<p class="para" id="fwk-redden-ch05_s05_s02_p16">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1390" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>9</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s02_p18"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1391" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>9</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>9</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>f</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>n</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mtext> </mtext><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>3</mn><mn>2</mn></msup><msup><mi>x</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>W</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mn>9</mn><mtext> </mtext><mi>a</mi><mi>s</mi><mtext> </mtext><msup><mn>3</mn><mn>2</mn></msup><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>a</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><msup><mn>3</mn><mrow><mn>2</mn><mo>(</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></msup><msup><mi>x</mi><mrow><mn>4</mn><mo>(</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><msup><mn>3</mn><mn>3</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>6</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mn>27</mn><msup><mi>x</mi><mn>6</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch05_s05_s02_p19">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1392" display="inline"><mrow><mfrac><mn>1</mn><mrow><mn>27</mn><msup><mi>x</mi><mn>6</mn></msup></mrow></mfrac></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s02_n05a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s05_s02_p20"><strong class="emphasis bold">Try this!</strong> Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1393" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>125</mn><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><msup><mi>b</mi><mn>6</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></mfrac></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch05_s05_s02_p21">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1394" display="inline"><mrow><mn>25</mn><msup><mi>b</mi><mn>4</mn></msup></mrow></math></span></p>
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</div>
<div class="section" id="fwk-redden-ch05_s05_s03" version="5.0" lang="en">
<h2 class="title editable block">Radical Expressions with Different Indices</h2>
<p class="para editable block" id="fwk-redden-ch05_s05_s03_p01">To apply the product or quotient rule for radicals, the indices of the radicals involved must be the same. If the indices are different, then first rewrite the radicals in exponential form and then apply the rules for exponents.</p>
<div class="callout block" id="fwk-redden-ch05_s05_s03_n01">
<h3 class="title">Example 14</h3>
<p class="para" id="fwk-redden-ch05_s05_s03_p02">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1395" display="inline"><mrow><msqrt><mn>2</mn></msqrt><mo>⋅</mo><mroot><mn>2</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s03_p03">In this example, the index of each radical factor is different. Hence the product rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents. Then apply the product rule for exponents.</p>
<p class="para" id="fwk-redden-ch05_s05_s03_p04"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1396" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>2</mn></msqrt><mo>⋅</mo><mroot><mn>2</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mtext> </mtext><mi>u</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>d</mi><mi>u</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>2</mn><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s05_s03_p05">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1397" display="inline"><mrow><mroot><mrow><msup><mn>2</mn><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s03_n02">
<h3 class="title">Example 15</h3>
<p class="para" id="fwk-redden-ch05_s05_s03_p06">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1398" display="inline"><mrow><mfrac><mrow><mroot><mn>4</mn><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow><mrow><mroot><mn>2</mn><mpadded width="0.6em"><mn>5</mn></mpadded></mroot></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s03_p07">In this example, the index of the radical in the numerator is different from the index of the radical in the denominator. Hence the quotient rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents and then apply the quotient rule for exponents.</p>
<p class="para" id="fwk-redden-ch05_s05_s03_p08"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1399" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mroot><mn>4</mn><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow><mrow><mroot><mn>2</mn><mpadded width="0.6em"><mn>5</mn></mpadded></mroot></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mroot><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow><mrow><mroot><mn>2</mn><mpadded width="0.6em"><mn>5</mn></mpadded></mroot></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mtext> </mtext><mi>u</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn><mo>−</mo><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mrow><mn>7</mn><mo>/</mo><mrow><mn>15</mn></mrow></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>2</mn><mn>7</mn></msup></mrow><mrow><mn>15</mn></mrow></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch05_s05_s03_p09">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1400" display="inline"><mrow><mroot><mrow><msup><mn>2</mn><mn>7</mn></msup></mrow><mrow><mn>15</mn></mrow></mroot></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s05_s03_n03">
<h3 class="title">Example 16</h3>
<p class="para" id="fwk-redden-ch05_s05_s03_p10">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1401" display="inline"><mrow><msqrt><mrow><mtext> </mtext><mroot><mn>4</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mtext> </mtext></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s05_s03_p11">Here the radicand of the square root is a cube root. After rewriting this expression using rational exponents, we will see that the power rule for exponents applies.</p>
<p class="para" id="fwk-redden-ch05_s05_s03_p12"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1402" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mtext> </mtext><mroot><mn>4</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mtext> </mtext></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mtext> </mtext><mroot><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mtext> </mtext></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mtext> </mtext><mi>u</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo>)</mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mn>2</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s05_s03_p13">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1403" display="inline"><mrow><mroot><mn>2</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
<div class="key_takeaways block" id="fwk-redden-ch05_s05_s03_n04">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch05_s05_s03_l01" mark="bullet">
<li>Any radical expression can be written in exponential form: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1404" display="inline"><mrow><mroot><mrow><msup><mi>a</mi><mi>m</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot><mo>=</mo><msup><mi>a</mi><mrow><mi>m</mi><mo>/</mo><mi>n</mi></mrow></msup></mrow><mo>.</mo></math></span>
</li>
<li>Fractional exponents indicate radicals. Use the numerator as the power and the denominator as the index of the radical.</li>
<li>All the rules of exponents apply to expressions with rational exponents.</li>
<li>If operations are to be applied to radicals with different indices, first rewrite the radicals in exponential form and then apply the rules for exponents.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch05_s05_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd01">
<h3 class="title">Part A: Rational Exponents</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch05_s05_qs01_p01"><strong class="emphasis bold">Express using rational exponents.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1405" display="inline"><mrow><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1407" display="inline"><mrow><msqrt><mn>6</mn></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1409" display="inline"><mrow><mroot><mn>3</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1411" display="inline"><mrow><mroot><mn>5</mn><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1413" display="inline"><mrow><mroot><mrow><msup><mn>5</mn><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1415" display="inline"><mrow><mroot><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1417" display="inline"><mrow><mroot><mrow><mn>49</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1419" display="inline"><mrow><mroot><mn>9</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1421" display="inline"><mrow><mroot><mi>x</mi><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1423" display="inline"><mrow><mroot><mi>x</mi><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1425" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>7</mn></msup></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1427" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa13">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1429" display="block"><mrow><mfrac><mn>1</mn><mrow><mpadded height="1.4em"><msqrt><mi>x</mi></msqrt></mpadded></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa14">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1431" display="block"><mrow><mfrac><mn>1</mn><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd01_qd02" start="15">
<p class="para" id="fwk-redden-ch05_s05_qs01_p30"><strong class="emphasis bold">Express in radical form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1433" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1435" display="inline"><mrow><msup><mrow><mn>11</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1437" display="inline"><mrow><msup><mn>7</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1439" display="inline"><mrow><msup><mn>2</mn><mrow><mn>3</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1441" display="inline"><mrow><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p41"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1443" display="inline"><mrow><msup><mi>x</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p43"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1445" display="inline"><mrow><msup><mi>x</mi><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1447" display="inline"><mrow><msup><mi>x</mi><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa23">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1449" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa24">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1451" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1453" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1455" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd01_qd03" start="27">
<p class="para" id="fwk-redden-ch05_s05_qs01_p55"><strong class="emphasis bold">Write as a radical and then simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1457" display="inline"><mrow><msup><mrow><mn>64</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1458" display="inline"><mrow><msup><mrow><mn>49</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa29">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1459" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa30">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1461" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>4</mn><mn>9</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p64"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1463" display="inline"><mrow><msup><mn>4</mn><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1465" display="inline"><mrow><msup><mn>9</mn><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa33">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1467" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa34">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1468" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p72"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1469" display="inline"><mrow><msup><mn>8</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p74"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1470" display="inline"><mrow><msup><mrow><mn>125</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa37">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1471" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>27</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa38">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1473" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>8</mn><mrow><mn>125</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p80"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1475" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>27</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p82"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1476" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>64</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p84"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1477" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p86"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1478" display="inline"><mrow><msup><mrow><mn>625</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p88"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1479" display="inline"><mrow><msup><mrow><mn>81</mn></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1481" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p92"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1483" display="inline"><mrow><mn>100</mn><mo>,</mo><msup><mrow><mn>000</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1484" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>32</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa47">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1485" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>32</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa48">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1487" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>243</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1489" display="inline"><mrow><msup><mn>9</mn><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1490" display="inline"><mrow><msup><mn>4</mn><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1491" display="inline"><mrow><msup><mn>8</mn><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1492" display="inline"><mrow><msup><mrow><mn>27</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1493" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1494" display="inline"><mrow><msup><mrow><mn>32</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa55">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1495" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa56">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1497" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>81</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1499" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>27</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1500" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>27</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1501" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>32</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1502" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>32</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd01_qd04" start="61">
<p class="para" id="fwk-redden-ch05_s05_qs01_p124"><strong class="emphasis bold">Use a calculator to approximate an answer rounded to the nearest hundredth.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p125"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1503" display="inline"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p127"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1504" display="inline"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p129"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1505" display="inline"><mrow><msup><mn>2</mn><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1506" display="inline"><mrow><msup><mn>3</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p133"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1507" display="inline"><mrow><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p135"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1508" display="inline"><mrow><msup><mn>7</mn><mrow><mn>1</mn><mo>/</mo><mn>7</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1509" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1510" display="inline"><mrow><mo>−</mo><msup><mn>9</mn><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p141">Explain why (−4)^(3/2) gives an error on a calculator and −4^(3/2) gives an answer of −8.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p142">Marcy received a text message from Mark asking her age. In response, Marcy texted back “125^(2/3) years old.” Help Mark determine Marcy’s age.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd02">
<h3 class="title">Part B: Operations Using the Rules of Exponents</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd02_qd01" start="71">
<p class="para" id="fwk-redden-ch05_s05_qs01_p144"><strong class="emphasis bold">Perform the operations and simplify. Leave answers in exponential form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p145"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1511" display="inline"><mrow><msup><mn>5</mn><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1512" display="inline"><mrow><msup><mn>3</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⋅</mo><msup><mn>3</mn><mrow><mn>7</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1513" display="inline"><mrow><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mn>5</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p151"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1515" display="inline"><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup><mo>⋅</mo><msup><mn>2</mn><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p153"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1517" display="inline"><mrow><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>⋅</mo><msup><mi>y</mi><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p155"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1519" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa77">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1521" display="block"><mrow><mfrac><mrow><msup><mn>5</mn><mrow><mn>11</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mn>5</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa78">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1522" display="block"><mrow><mfrac><mrow><msup><mn>2</mn><mrow><mn>9</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa79">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1523" display="block"><mrow><mfrac><mrow><mn>2</mn><msup><mi>a</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa80">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1525" display="block"><mrow><mfrac><mrow><mn>3</mn><msup><mi>b</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>b</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa81">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1527" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>8</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa82">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1528" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>3</mn><mn>6</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa83">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1529" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa84">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1531" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa85">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1533" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>8</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa86">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1535" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>6</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa87">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1537" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa88">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1539" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>9</mn><msup><mi>x</mi><mn>6</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa89">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1541" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa90">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1543" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>8</mn><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa91">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1545" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>36</mn><msup><mi>x</mi><mn>4</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa92">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1547" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>6</mn></msup><msup><mi>z</mi><mrow><mtext>−</mtext><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa93">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1549" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mi>a</mi><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow><mrow><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa94">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1551" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mi>b</mi><mrow><mn>4</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow><mrow><msup><mi>b</mi><mrow><mn>1</mn><mo>/</mo><mn>10</mn></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>10</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa95">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1553" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa96">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1555" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>27</mn><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow><mrow><msup><mi>y</mi><mn>9</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa97">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1557" display="block"><mrow><mfrac><mrow><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa98">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1559" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>10</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa99">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1561" display="block"><mrow><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa100">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1563" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mn>5</mn><mo>/</mo><mn>4</mn></mrow></msup><mi>y</mi></mrow><mrow><mi>x</mi><msup><mi>y</mi><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa101">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1565" display="block"><mrow><mfrac><mrow><mn>49</mn><msup><mi>a</mi><mrow><mn>5</mn><mo>/</mo><mn>7</mn></mrow></msup><msup><mi>b</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><mn>7</mn><msup><mi>a</mi><mrow><mn>3</mn><mo>/</mo><mn>7</mn></mrow></msup><msup><mi>b</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa102">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1567" display="block"><mrow><mfrac><mrow><mn>16</mn><msup><mi>a</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup><msup><mi>b</mi><mrow><mn>5</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow><mrow><mn>8</mn><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>b</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa103">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1569" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>9</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mn>6</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>y</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa104">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1571" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>125</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mrow><mn>3</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><mi>x</mi><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa105">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1573" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>27</mn><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><msup><mi>b</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup><msup><mi>b</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa106">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1575" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>25</mn><msup><mi>a</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mi>b</mi><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup><msup><mi>b</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p217"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1577" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mi>z</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p219"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1579" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>81</mn><msup><mi>x</mi><mn>8</mn></msup><msup><mi>y</mi><mrow><mtext>−</mtext><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mi>z</mi><mrow><mtext>−</mtext><mn>4</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p221"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1581" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>100</mn><msup><mi>a</mi><mrow><mtext>−</mtext><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mi>b</mi><mn>4</mn></msup><msup><mi>c</mi><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p223"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1583" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>125</mn><msup><mi>a</mi><mn>9</mn></msup><msup><mi>b</mi><mrow><mtext>−</mtext><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup><msup><mi>c</mi><mrow><mtext>−</mtext><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mtext>−</mtext><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd03">
<h3 class="title">Part C: Radical Expressions with Different Indices</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd03_qd01" start="111">
<p class="para" id="fwk-redden-ch05_s05_qs01_p225"><strong class="emphasis bold">Perform the operations.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p226"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1585" display="inline"><mrow><mroot><mn>9</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>⋅</mo><mroot><mn>3</mn><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p228"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1587" display="inline"><mrow><msqrt><mn>5</mn></msqrt><mo>⋅</mo><mroot><mrow><mn>25</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa113">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p230"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1589" display="inline"><mrow><msqrt><mi>x</mi></msqrt><mo>⋅</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa114">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p232"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1591" display="inline"><mrow><msqrt><mi>y</mi></msqrt><mo>⋅</mo><mroot><mi>y</mi><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa115">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p234"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1593" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>⋅</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa116">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p236"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1595" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>⋅</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa117">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1597" display="block"><mrow><mfrac><mrow><mroot><mrow><mn>100</mn></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow><mrow><mpadded height="1.4em"><msqrt><mrow><mn>10</mn></mrow></msqrt></mpadded></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa118">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1599" display="block"><mrow><mfrac><mrow><mroot><mrow><mn>16</mn></mrow><mpadded width="0.6em"><mn>5</mn></mpadded></mroot></mrow><mrow><mroot><mn>4</mn><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa119">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1601" display="block"><mrow><mfrac><mrow><mroot><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow><mrow><mpadded height="1.4em"><msqrt><mi>a</mi></msqrt></mpadded></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa120">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1603" display="block"><mrow><mfrac><mrow><mroot><mrow><msup><mi>b</mi><mn>4</mn></msup></mrow><mpadded width="0.6em"><mn>5</mn></mpadded></mroot></mrow><mrow><mroot><mi>b</mi><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa121">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1605" display="block"><mrow><mfrac><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow><mrow><mroot><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mpadded width="0.6em"><mn>5</mn></mpadded></mroot></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa122">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1607" display="block"><mrow><mfrac><mrow><mroot><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mpadded width="0.6em"><mn>4</mn></mpadded></mroot></mrow><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa123">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p250"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1609" display="inline"><mrow><msqrt><mrow><mtext> </mtext><mroot><mrow><mn>16</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa124">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p252"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1611" display="inline"><mrow><msqrt><mrow><mtext> </mtext><mroot><mn>9</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa125">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p254"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1613" display="inline"><mrow><mroot><mrow><mtext> </mtext><mroot><mn>2</mn><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa126">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p256"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1615" display="inline"><mrow><mroot><mrow><mtext> </mtext><mroot><mn>5</mn><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa127">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p258"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1617" display="inline"><mrow><mroot><mrow><mtext> </mtext><mpadded height="1.4em"><msqrt><mn>7</mn></msqrt></mpadded></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa128">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p260"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1619" display="inline"><mrow><mroot><mrow><mtext> </mtext><mpadded height="1.4em"><msqrt><mn>3</mn></msqrt></mpadded></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd04">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s05_qs01_qd04_qd01" start="129">
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa129">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p262">Who is credited for devising the notation that allows for rational exponents? What are some of his other accomplishments?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa130">
<div class="question">
<p class="para" id="fwk-redden-ch05_s05_qs01_p263">When using text, it is best to communicate <em class="emphasis">n</em>th roots using rational exponents. Give an example.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch05_s05_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch05_s05_qs01_p03_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1406" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s05_qs01_qa02_ans" audience="instructoronly">