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<div class="section" id="fwk-redden-ch01_s05" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">1.5</span> Rules of Exponents and Scientific Notation</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch01_s05_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch01_s05_o01" numeration="arabic">
<li>Review the rules of exponents.</li>
<li>Review the definition of negative exponents and zero as an exponent.</li>
<li>Work with numbers using scientific notation.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch01_s05_s01" version="5.0" lang="en">
<h2 class="title editable block">Review of the Rules of Exponents</h2>
<p class="para block" id="fwk-redden-ch01_s05_s01_p01">In this section, we review the rules of exponents. Recall that if a factor is repeated multiple times, then the product can be written in exponential form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1021" display="inline"><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow><mo>.</mo></math></span> The positive integer exponent <em class="emphasis">n</em> indicates the number of times the base <em class="emphasis">x</em> is repeated as a factor.</p>
<div class="informalfigure large block">
<img src="section_04/941a37ca5e0918d9f71ed52a9404982e.png">
</div>
<p class="para block" id="fwk-redden-ch01_s05_s01_p03">Consider the product of <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1022" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1023" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup></mrow></math></span>,</p>
<div class="informalfigure large block">
<img src="section_04/0e0964df4e8cac885e6c6af89e2b4d36.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p05">Expanding the expression using the definition produces multiple factors of the base which is quite cumbersome, particularly when <em class="emphasis">n</em> is large. For this reason, we have useful rules to help us simplify expressions with exponents. In this example, notice that we could obtain the same result by adding the exponents.</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1024" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>6</mn></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>4</mn><mo>+</mo><mn>6</mn></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>10</mn></mrow></msup></mrow></mtd><mtd><mrow><mstyle color="#007fbf"><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></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p07">In general, this describes the <span class="margin_term"><a class="glossterm">product rule for exponents</a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1025" 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>; the product of two expressions with the same base can be simplified by adding the exponents.</span></span>. In other words, when multiplying two expressions with the same base we add the exponents. Compare this to raising a factor involving an exponent to a power, such as <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1026" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>6</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large block">
<img src="section_04/e7648909f847c0d277431ad91629a867.png">
</div>
<p class="para block" id="fwk-redden-ch01_s05_s01_p09">Here we have 4 factors of <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1027" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup></mrow></math></span>, which is equivalent to multiplying the exponents.</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1028" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>6</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>6</mn><mo>⋅</mo><mn>4</mn></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>24</mn></mrow></msup></mrow></mtd><mtd><mrow><mstyle color="#007fbf"><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></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p11">This describes the <span class="margin_term"><a class="glossterm">power rule for exponents</a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1029" 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><mi>n</mi></mrow></msup></mrow></math></span>; a power raised to a power can be simplified by multiplying the exponents.</span></span>. Now we consider raising grouped products to a power. For example,</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1030" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow><mn>4</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mi> </mi><mo>⋅</mo><mi> </mi><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mi> </mi><mo>⋅</mo><mi> </mi><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mi> </mi><mo>⋅</mo><mi> </mi><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>2</mn></msup><mi> </mi><mo>⋅</mo><mi> </mi><msup><mi>y</mi><mn>3</mn></msup><mo>⋅</mo><msup><mi>y</mi><mn>3</mn></msup><mo>⋅</mo><msup><mi>y</mi><mn>3</mn></msup><mo>⋅</mo><msup><mi>y</mi><mn>3</mn></msup><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>p</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>y</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mrow><mn>2</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mi>y</mi><mrow><mn>3</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>3</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>8</mn></msup><msup><mi>y</mi><mrow><mn>12</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p13">After expanding, we are left with four factors of the product <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1031" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mo>.</mo></math></span> This is equivalent to raising each of the original grouped factors to the fourth power and applying the power rule.</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1032" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup><mo>=</mo><msup><mi>x</mi><mn>8</mn></msup><msup><mi>y</mi><mrow><mn>12</mn></mrow></msup></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p15">In general, this describes the use of the power rule for a product as well as the power rule for exponents. In summary, the rules of exponents streamline the process of working with algebraic expressions and will be used extensively as we move through our study of algebra. Given any positive integers <em class="emphasis">m</em> and <em class="emphasis">n</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1033" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>≠</mo><mn>0</mn></mrow></math></span> we have</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p16">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Product rule for exponents:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1034" 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="right"><p class="para"><strong class="emphasis bold">Quotient rule for exponents:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1035" 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></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Power rule for exponents:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1036" 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="right"><p class="para"><strong class="emphasis bold">Power rule for a product</strong>:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1037" 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="right"><p class="para"><strong class="emphasis bold">Power rule for a quotient</strong>:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1038" 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></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1039" 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>; if a product is raised to a power, then apply that power to each factor in the product.</span></span></p>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1040" 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><mi> </mi></mrow></math></span>; if a quotient is raised to a power, then apply that power to the numerator and the denominator.</span></span></p>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p17">These rules allow us to efficiently perform operations with exponents.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p18">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1041" display="inline"><mrow><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p19"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1042" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup></mrow></mfrac><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><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></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mn>10</mn><mrow><mn>16</mn><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><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></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mn>10</mn><mrow><mn>13</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p20">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1043" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>13</mn></mrow></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p21">In the previous example, notice that we did not multiply the base 10 times itself. When applying the product rule, add the exponents and leave the base unchanged.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p22">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1044" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>5</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>4</mn></msup><mo>⋅</mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p23">Recall that the variable <em class="emphasis">x</em> is assumed to have an exponent of one, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1045" display="inline"><mrow><mi>x</mi><mo>=</mo><msup><mi>x</mi><mn>1</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p24"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1046" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>5</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>4</mn></msup><mo>⋅</mo><mi>x</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>5</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>10</mn></mrow></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mrow><mn>10</mn><mo>⋅</mo><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mrow><mn>20</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p25">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1047" display="inline"><mrow><msup><mi>x</mi><mrow><mn>20</mn></mrow></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p26">The base could in fact be any algebraic expression.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p27">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1048" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>9</mn></msup><mi> </mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>13</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p28">Treat the expression <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1049" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> as the base.</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1050" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mn>9</mn></msup><mi> </mi><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mn>13</mn></mrow></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mn>9</mn><mo>+</mo><mn>13</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mn>22</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p30">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1051" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>22</mn></mrow></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p31">The commutative property of multiplication allows us to use the product rule for exponents to simplify factors of an algebraic expression.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p32">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1052" display="inline"><mrow><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>5</mn></msup><mi>y</mi><mi> </mi><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mtext> </mtext><mn>3</mn><msup><mi>x</mi><mn>7</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p33">Multiply the coefficients and add the exponents of variable factors with the same base.</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p34"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1053" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>5</mn></msup><mi>y</mi><mi> </mi><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mtext> </mtext><mn>3</mn><msup><mi>x</mi><mn>7</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn><mo>⋅</mo><mn>3</mn><mi> </mi><mi> </mi><mo>⋅</mo><mi> </mi><msup><mi>x</mi><mn>5</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>7</mn></msup><mi> </mi><mi> </mi><mo>⋅</mo><mi> </mi><mi> </mi><msup><mi>y</mi><mn>1</mn></msup><mo>⋅</mo><msup><mi>y</mi><mn>3</mn></msup><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>p</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>y</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi> </mi><mi> </mi><mo>−</mo><mn>24</mn><mi> </mi><mo>⋅</mo><mi> </mi><msup><mi>x</mi><mrow><mn>5</mn><mo>+</mo><mn>7</mn></mrow></msup><mi> </mi><mi> </mi><mo>⋅</mo><mi> </mi><mi> </mi><msup><mi>y</mi><mrow><mn>1</mn><mo>+</mo><mn>3</mn></mrow></msup><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><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></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>24</mn><msup><mi>x</mi><mrow><mn>12</mn></mrow></msup><msup><mi>y</mi><mn>4</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p35">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1054" display="inline"><mrow><mo>−</mo><mn>24</mn><msup><mi>x</mi><mrow><mn>12</mn></mrow></msup><msup><mi>y</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p36">Division involves the quotient rule for exponents.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p37">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1055" display="inline"><mrow><mfrac><mrow><mn>33</mn><msup><mi>x</mi><mn>7</mn></msup><msup><mi>y</mi><mn>5</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>10</mn></mrow></msup></mrow><mrow><mn>11</mn><msup><mi>x</mi><mn>6</mn></msup><mi>y</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p38"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1056" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>33</mn><msup><mi>x</mi><mn>7</mn></msup><msup><mi>y</mi><mn>5</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>10</mn></mrow></msup></mrow><mrow><mn>11</mn><msup><mi>x</mi><mn>6</mn></msup><mi>y</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>33</mn></mrow><mrow><mn>11</mn></mrow></mfrac><mi> </mi><mi> </mi><mo>⋅</mo><mi> </mi><mi> </mi><msup><mi>x</mi><mrow><mn>7</mn><mo>−</mo><mn>6</mn></mrow></msup><mi> </mi><mo>⋅</mo><msup><mi>y</mi><mrow><mn>5</mn><mo>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mn>10</mn><mo>−</mo><mn>3</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><msup><mi>x</mi><mn>1</mn></msup><msup><mi>y</mi><mn>4</mn></msup><mi> </mi><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow><mn>7</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p39">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1057" display="inline"><mrow><mn>3</mn><mi>x</mi><msup><mi>y</mi><mn>4</mn></msup><mi> </mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>7</mn></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p40">The power rule for a quotient allows us to apply that exponent to the numerator and denominator. This rule requires that the denominator is nonzero and so we will make this assumption for the remainder of the section.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n06">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p41">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1058" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi></mrow><mrow><msup><mi>c</mi><mn>4</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p42">First apply the power rule for a quotient and then the power rule for a product.</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p43"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1059" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi></mrow><mrow><msup><mi>c</mi><mn>4</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mn>3</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>c</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mfrac><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><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>q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>c</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mfrac><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><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></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>64</mn><msup><mi>a</mi><mn>6</mn></msup><msup><mi>b</mi><mn>3</mn></msup></mrow><mrow><msup><mi>c</mi><mrow><mn>12</mn></mrow></msup></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p44">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1060" display="inline"><mrow><mo>−</mo><mfrac><mrow><mn>64</mn><msup><mi>a</mi><mn>6</mn></msup><msup><mi>b</mi><mn>3</mn></msup></mrow><mrow><msup><mi>c</mi><mrow><mn>12</mn></mrow></msup></mrow></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p45">Using the quotient rule for exponents, we can define what it means to have zero as an exponent. Consider the following calculation:</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p46"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1061" display="block"><mrow><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>=</mo><mfrac><mrow><mn>25</mn></mrow><mrow><mn>25</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mn>5</mn><mn>2</mn></msup></mrow><mrow><msup><mn>5</mn><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mn>5</mn><mrow><mn>2</mn><mo>−</mo><mn>2</mn></mrow></msup><mo>=</mo><mstyle color="#007fbf"><msup><mn>5</mn><mn>0</mn></msup></mstyle></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p47">Twenty-five divided by twenty-five is clearly equal to one, and when the quotient rule for exponents is applied, we see that a zero exponent results. In general, given any nonzero real number <em class="emphasis">x</em> and integer <em class="emphasis">n</em>,</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p48"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1062" display="block"><mrow><mn>1</mn><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mi>n</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mn>0</mn></msup></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p49">This leads us to the definition of <span class="margin_term"><a class="glossterm">zero as an exponent</a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1063" display="inline"><mrow><msup><mi>x</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn><mi> </mi></mrow></math></span>; any nonzero base raised to the 0 power is defined to be 1.</span></span>,</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p50"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1064" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><msup><mi>x</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p51">It is important to note that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1065" display="inline"><mrow><msup><mn>0</mn><mn>0</mn></msup></mrow></math></span> is indeterminate. If the base is negative, then the result is still positive one. In other words, any nonzero base raised to the zero power is defined to be equal to one. In the following examples assume all variables are nonzero.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n07">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p52">Simplify:</p>
<ol class="orderedlist" id="fwk-redden-ch01_s05_s01_o01" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1066" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>0</mn></msup></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1067" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>0</mn></msup></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch01_s05_s01_o02" numeration="loweralpha">
<li>
<p class="para">Any nonzero quantity raised to the zero power is equal to 1.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1068" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</li>
<li>
<p class="para">In the example, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1069" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>0</mn></msup></mrow></math></span>, the base is <em class="emphasis">x</em>, not −2<em class="emphasis">x</em>.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1070" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>0</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mo>⋅</mo><msup><mi>x</mi><mn>0</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math></span></p>
</li>
</ol>
</div>
<p class="para block" id="fwk-redden-ch01_s05_s01_p53">Noting that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1071" display="inline"><mrow><msup><mn>2</mn><mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span> we can write,</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p54"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1072" display="block"><mrow><mstyle color="#007fbf"><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow></mfrac></mstyle><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mn>0</mn></msup></mrow><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow></mfrac><mo>=</mo><msup><mn>2</mn><mrow><mn>0</mn><mo>−</mo><mn>3</mn></mrow></msup><mo>=</mo><mstyle color="#007fbf"><msup><mn>2</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mstyle></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p55">In general, given any nonzero real number <em class="emphasis">x</em> and integer <em class="emphasis">n</em>,</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p56"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1073" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>0</mn></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mn>0</mn><mo>−</mo><mi>n</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p57">This leads us to the definition of <span class="margin_term"><a class="glossterm">negative exponents</a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1074" 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><mi> </mi></mrow></math></span>, given any integer <em class="emphasis">n</em>, where <em class="emphasis">x</em> is nonzero.</span></span>:</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p58"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1075" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><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><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p59">An expression is completely simplified if it does not contain any negative exponents.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n08">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p60">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1076" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p61">Rewrite the entire quantity in the denominator with an exponent of 2 and then simplify further.</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p62"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1077" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mi> </mi><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mi> </mi><msup><mrow><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mrow><mn>16</mn><msup><mi>x</mi><mn>4</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p63">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1078" display="inline"><mrow><mfrac><mn>1</mn><mrow><mn>16</mn><msup><mi>x</mi><mn>4</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p64">Sometimes negative exponents appear in the denominator.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n09">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p65">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1079" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p66"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1080" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow></mfrac></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p67">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1081" display="inline"><mrow><mfrac><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch01_s05_s01_p68">The previous example suggests a property of <span class="margin_term"><a class="glossterm">quotients with negative exponents</a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1082" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mo>−</mo><mi>m</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mi>m</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mi> </mi></mrow></math></span>, given any integers <em class="emphasis">m</em> and <em class="emphasis">n</em>, where <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1083" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn><mi> </mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1084" display="inline"><mrow><mi>y</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span>. Given any integers <em class="emphasis">m</em> and <em class="emphasis">n</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1085" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn><mi> </mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1086" display="inline"><mrow><mi>y</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, then</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p69"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1087" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mo>−</mo><mi>m</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mtext> </mtext><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mi>m</mi></msup></mrow></mfrac></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mi>y</mi><mi>m</mi></msup></mrow><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mi>m</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p70">This leads us to the property</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p71"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1088" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mo>−</mo><mi>m</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mi>m</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mi> </mi></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p72">In other words, negative exponents in the numerator can be written as positive exponents in the denominator and negative exponents in the denominator can be written as positive exponents in the numerator.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n10">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p73">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1089" display="inline"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mrow><msup><mi>z</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p74">Take care with the coefficient −5, recognize that this is the base and that the exponent is actually positive one: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1090" display="inline"><mrow><mo>−</mo><mn>5</mn><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>1</mn></msup></mrow><mo>.</mo></math></span> Hence, the rules of negative exponents do not apply to this coefficient; leave it in the numerator.</p>
<p class="para" id="fwk-redden-ch01_s05_s01_p75"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1091" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mrow><msup><mi>z</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>5</mn><mi> </mi><mi> </mi><mstyle color="#007fbf"><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mstyle><mi> </mi><mi> </mi><msup><mi>y</mi><mn>3</mn></msup></mrow><mrow><mstyle color="#007f3f"><msup><mi>z</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mstyle></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>5</mn><mi> </mi><msup><mi>y</mi><mn>3</mn></msup><mi> </mi><mstyle color="#007f3f"><msup><mi>z</mi><mn>4</mn></msup></mstyle></mrow><mrow><mstyle color="#007fbf"><msup><mi>x</mi><mn>3</mn></msup></mstyle></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p76">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1092" display="inline"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>3</mn></msup><msup><mi>z</mi><mn>4</mn></msup></mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch01_s05_s01_p77">In summary, given integers <em class="emphasis">m</em> and <em class="emphasis">n</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1093" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>≠</mo><mn>0</mn></mrow></math></span> we have</p>
<p class="para block" id="fwk-redden-ch01_s05_s01_p78">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Zero exponent</strong>:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1094" display="inline"><mrow><msup><mi>x</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Negative exponent</strong>:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1095" 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="right"><p class="para"><strong class="emphasis bold">Quotients with negative exponents</strong>:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1096" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mo>−</mo><mi>m</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mi>m</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mi> </mi></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s01_p79">Furthermore, all of the rules of exponents defined so far extend to any integer exponents. We will expand the scope of these properties to include any real number exponents later in the course.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s01_n10a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch01_s05_s01_p80"><strong class="emphasis bold">Try this!</strong> Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1097" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mi>z</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s01_p81">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1098" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>8</mn></msup><msup><mi>z</mi><mn>4</mn></msup></mrow><mrow><mn>16</mn><msup><mi>y</mi><mrow><mn>12</mn></mrow></msup></mrow></mfrac></mrow></math></span></p>
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</div>
</div>
</div>
<div class="section" id="fwk-redden-ch01_s05_s02" version="5.0" lang="en">
<h2 class="title editable block">Scientific Notation</h2>
<p class="para block" id="fwk-redden-ch01_s05_s02_p01">Real numbers expressed using <span class="margin_term"><a class="glossterm">scientific notation</a><span class="glossdef">Real numbers expressed the form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1099" display="inline"><mrow><mi>a</mi><mo>×</mo><msup><mrow><mn>10</mn></mrow><mi>n</mi></msup></mrow></math></span>, where <em class="emphasis">n</em> is an integer and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1100" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn></mrow><mo>.</mo></math></span></span></span> have the form,
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1101" display="block"><mrow><mi>a</mi><mo>×</mo><msup><mrow><mn>10</mn></mrow><mi>n</mi></msup></mrow></math></span>
where <em class="emphasis">n</em> is an integer and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1102" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn></mrow><mo>.</mo></math></span> This form is particularly useful when the numbers are very large or very small. For example,</p>
<p class="para block" id="fwk-redden-ch01_s05_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1103" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>9,460,000,000,000,000</mn><mi> </mi><mtext>m</mtext></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9.46</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>15</mn></mrow></msup><mi> </mi><mtext>m</mtext><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>O</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>l</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi><mtext> </mtext><mi>y</mi><mi>e</mi><mi>a</mi><mi>r</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>0.000000000025</mn><mi> </mi><mtext>m</mtext></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2.5</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>11</mn></mrow></msup><mi> </mi><mtext>m</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>R</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>u</mi><mi>s</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mi>h</mi><mi>y</mi><mi>d</mi><mi>r</mi><mi>o</mi><mi>g</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>a</mi><mi>t</mi><mi>o</mi><mi>m</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s02_p05">It is cumbersome to write all the zeros in both of these cases. Scientific notation is an alternative, compact representation of these numbers. The factor <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1104" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mi>n</mi></msup></mrow></math></span> indicates the power of ten to multiply the coefficient by to convert back to decimal form:</p>
<div class="informalfigure large block">
<img src="section_04/ec3870354c3c40177bbcbc35cae4c248.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s02_p07">This is equivalent to moving the decimal in the coefficient fifteen places to the right.</p>
<p class="para editable block" id="fwk-redden-ch01_s05_s02_p08">A negative exponent indicates that the number is very small:</p>
<div class="informalfigure large block">
<img src="section_04/952aa4973b82a6fbb9ed079645ebee29.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s02_p09">This is equivalent to moving the decimal in the coefficient eleven places to the left.</p>
<p class="para block"> </p>
<p class="para block" id="fwk-redden-ch01_s05_s02_p10">Converting a decimal number to scientific notation involves moving the decimal as well. Consider all of the equivalent forms of <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1105" display="inline"><mrow><mn>0.00563</mn></mrow></math></span> with factors of 10 that follow:</p>
<p class="para block" id="fwk-redden-ch01_s05_s02_p11"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1106" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>0.00563</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0.0563</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0.563</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mstyle color="#007fbf"><mo>=</mo></mstyle></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>5.63</mn><mtext> </mtext><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>56.3</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>563</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch01_s05_s02_p12">While all of these are equal, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1107" display="inline"><mrow><mn>5.63</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math></span> is the only form expressed in correct scientific notation. This is because the coefficient 5.63 is between 1 and 10 as required by the definition. Notice that we can convert <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1108" display="inline"><mrow><mn>5.63</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math></span> back to decimal form, as a check, by moving the decimal three places to the left.</p>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1109" 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></mrow></math></span>; the quotient of two expressions with the same base can be simplified by subtracting the exponents.</span></span></p>
<div class="callout block" id="fwk-redden-ch01_s05_s02_n01">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch01_s05_s02_p13">Write <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1110" display="inline"><mrow><mn>1,075,000,000,000</mn></mrow></math></span> using scientific notation.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p14">Here we count twelve decimal places to the left of the decimal point to obtain the number 1.075.</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1111" display="block"><mrow><mn>1,075,000,000,000</mn><mo>=</mo><mn>1.075</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p16">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1112" display="inline"><mrow><mn>1.075</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s05_s02_n02">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch01_s05_s02_p17">Write <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1113" display="inline"><mrow><mn>0.000003045</mn></mrow></math></span> using scientific notation.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p18">Here we count six decimal places to the right to obtain 3.045.</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p19"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1114" display="block"><mrow><mn>0.000003045</mn><mo>=</mo><mn>3.045</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p20">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1115" display="inline"><mrow><mn>3.045</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s05_s02_p21">Often we will need to perform operations when using numbers in scientific notation. All the rules of exponents developed so far also apply to numbers in scientific notation.</p>
<div class="callout block" id="fwk-redden-ch01_s05_s02_n03">
<h3 class="title">Example 13</h3>
<p class="para" id="fwk-redden-ch01_s05_s02_p22">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1116" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4.36</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5.3</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p23">Use the fact that multiplication is commutative, and apply the product rule for exponents.</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p24"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1117" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>4.36</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5.30</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi> </mi><mrow><mo>(</mo><mrow><mn>4.36</mn><mo>⋅</mo><mn>5.30</mn></mrow><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>23.108</mn></mstyle><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn><mo>+</mo><mn>12</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>2.3108</mn><mo>×</mo><msup><mn>10</mn><mn>1</mn></msup></mstyle><mi> </mi><mo>×</mo><mi> </mi><mi> </mi><msup><mn>10</mn><mn>7</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2.3108</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>1</mn><mo>+</mo><mn>7</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2.3108</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p25">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1118" display="inline"><mrow><mn>2.3108</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s05_s02_n04">
<h3 class="title">Example 14</h3>
<p class="para" id="fwk-redden-ch01_s05_s02_p26">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1119" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3.24</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow><mo>)</mo></mrow><mo>÷</mo><mrow><mo>(</mo><mrow><mn>9.0</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p27"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1120" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mrow><mo>(</mo><mrow><mn>3.24</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>9.0</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mfrac><mrow><mn>3.24</mn></mrow><mrow><mn>9.0</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0.36</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>8</mn><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>0.36</mn></mstyle><mo>×</mo><msup><mn>10</mn><mrow><mn>8</mn><mo>+</mo><mn>3</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>3.6</mn><mo>×</mo></mstyle><msup><mstyle color="#007fbf"><mn>10</mn></mstyle><mrow><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle></mrow></msup><mo>×</mo><msup><mn>10</mn><mrow><mn>11</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3.6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mn>11</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3.6</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>10</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p28">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1121" display="inline"><mrow><mn>3.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>10</mn></mrow></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s05_s02_n05">
<h3 class="title">Example 15</h3>
<p class="para" id="fwk-redden-ch01_s05_s02_p29">The speed of light is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1122" display="inline"><mrow><mn>6.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow></math></span> miles per hour. Express this speed in miles per second.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p30">A unit analysis indicates that we must divide the number by 3,600.</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p31"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1123" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>6.7</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mi> </mi><mtext>miles</mtext><mtext> </mtext><mtext>per</mtext><mtext> </mtext><mtext>hour</mtext></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>6.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup><mi> </mi><mtext>miles</mtext></mrow><mrow><mn>1</mn><mi> </mi><mtext> </mtext><menclose notation="updiagonalstrike"><mrow><mstyle color="#ff0000"><mtext>hour</mtext></mstyle></mrow></menclose></mrow></mfrac><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>1</mn><mi> </mi><mtext> </mtext><menclose notation="updiagonalstrike"><mrow><mstyle color="#ff0000"><mtext>hour</mtext></mstyle></mrow></menclose></mrow><mrow><mn>60</mn><mi> </mi><menclose notation="updiagonalstrike"><mrow><mstyle color="#007f3f"><mtext>minutes</mtext></mstyle></mrow></menclose></mrow></mfrac></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>1</mn><mtext> </mtext><menclose notation="updiagonalstrike"><mrow><mstyle color="#007f3f"><mtext>minutes</mtext></mstyle></mrow></menclose></mrow><mrow><mn>60</mn><mi> </mi><mtext>seconds</mtext></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>6.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup><mi> </mi><mtext>miles</mtext></mrow><mrow><mn>3600</mn><mtext> </mtext><mtext>seconds</mtext></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mfrac><mrow><mn>6.7</mn></mrow><mrow><mn>3600</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mi> </mi><mi> </mi></mtd></mtr><mtr><mtd columnalign="left"></mtd><mtd><mo>≈</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>0.0019</mn></mstyle><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mtext> </mtext><mstyle color="#007fbf"><mi>r</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mi> </mi><mi>t</mi><mi>o</mi><mi> </mi><mi>t</mi><mi>w</mi><mi>o</mi><mi> </mi><mi>s</mi><mi>i</mi><mi>g</mi><mi>n</mi><mi>i</mi><mi>f</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>n</mi><mi>t</mi><mi> </mi><mi>d</mi><mi>i</mi><mi>g</mi><mi>i</mi><mi>t</mi><mi>s</mi></mstyle></mtd><mtd columnalign="left"></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>1.9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mstyle><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mi> </mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1.9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn><mo>+</mo><mn>8</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1.9</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p32">Answer: The speed of light is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1124" display="inline"><mrow><mn>1.9</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>5</mn></msup></mrow></math></span> miles per second.</p>
</div>
<div class="callout block" id="fwk-redden-ch01_s05_s02_n06">
<h3 class="title">Example 16</h3>
<p class="para" id="fwk-redden-ch01_s05_s02_p33">The Sun moves around the center of the galaxy in a nearly circular orbit. The distance from the center of our galaxy to the Sun is approximately 26,000 light-years. What is the circumference of the orbit of the Sun around the galaxy in meters?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p34">One light-year measures <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1125" display="inline"><mrow><mn>9.46</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>15</mn></mrow></msup></mrow></math></span> meters. Therefore, multiply this by 26,000 or <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1126" display="inline"><mrow><mn>2.60</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup></mrow></math></span> to find the length of 26,000 light years in meters.</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p35"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1127" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>9.46</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>15</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2.60</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9.46</mn><mo>⋅</mo><mn>2.60</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>15</mn></mrow></msup><mo>⋅</mo><msup><mn>10</mn><mn>4</mn></msup></mtd></mtr><mtr><mtd columnalign="left"></mtd><mtd><mo>≈</mo></mtd><mtd columnalign="left"><mn>24.6</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>19</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2.46</mn><mo>×</mo><msup><mn>10</mn><mn>1</mn></msup><mo>⋅</mo><msup><mn>10</mn><mrow><mn>19</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2.46</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>20</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p36">The radius <em class="emphasis">r</em> of this very large circle is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1128" display="inline"><mrow><mn>2.46</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>20</mn></mrow></msup></mrow></math></span> meters. Use the formula <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1129" display="inline"><mrow><mi>C</mi><mo>=</mo><mn>2</mn><mi mathvariant="italic">π</mi><mi>r</mi></mrow></math></span> to calculate the circumference of the orbit.</p>
<p class="para" id="fwk-redden-ch01_s05_s02_p37"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1130" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>C</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mi mathvariant="italic">π</mi><mi>r</mi></mtd></mtr><mtr><mtd columnalign="left"></mtd><mtd columnalign="left"><mo>≈</mo></mtd><mtd columnalign="left"><mn>2</mn><mrow><mo>(</mo><mrow><mn>3.14</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2.46</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>20</mn></mrow></msup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15.4</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>20</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1.54</mn><mo>×</mo><msup><mn>10</mn><mn>1</mn></msup><mo>⋅</mo><msup><mn>10</mn><mrow><mn>20</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1.54</mn><mo>×</mo><msup><mn>10</mn><mrow><mn>21</mn></mrow></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p38">Answer: The circumference of the Sun’s orbit is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1131" display="inline"><mrow><mn>1.54</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>21</mn></mrow></msup></mrow></math></span> meters.</p>
</div>
<div class="callout block" id="fwk-redden-ch01_s05_s02_n06a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch01_s05_s02_p39"><strong class="emphasis bold">Try this!</strong> Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1132" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3.15</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><mo>)</mo></mrow><mo>÷</mo><mrow><mo>(</mo><mrow><mn>12</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>13</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s05_s02_p40">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1133" display="inline"><mrow><mn>2.625</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>7</mn></msup></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/jOiRSs7hyW4" condition="http://img.youtube.com/vi/jOiRSs7hyW4/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/jOiRSs7hyW4" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch01_s05_s02_n07">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch01_s05_s02_l01" mark="bullet">
<li>When multiplying two quantities with the same base, add exponents: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1134" 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><mo>.</mo></math></span>
</li>
<li>When dividing two quantities with the same base, subtract exponents: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1135" 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></mrow><mo>.</mo></math></span>
</li>
<li>When raising powers to powers, multiply exponents: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1136" 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><mo>.</mo></math></span>
</li>
<li>When a grouped quantity involving multiplication and division is raised to a power, apply that power to all of the factors in the numerator and the denominator: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1137" 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> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1138" 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><mi> </mi></mrow><mo>.</mo></math></span>
</li>
<li>Any nonzero quantity raised to the 0 power is defined to be equal to 1: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1139" display="inline"><mrow><msup><mi>x</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn><mi> </mi></mrow><mo>.</mo></math></span>
</li>
<li>Expressions with negative exponents in the numerator can be rewritten as expressions with positive exponents in the denominator: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1140" 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><mi> </mi></mrow><mo>.</mo></math></span>
</li>
<li>Expressions with negative exponents in the denominator can be rewritten as expressions with positive exponents in the numerator: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1141" display="inline"><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>m</mi></mrow></msup></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mi>m</mi></msup><mi> </mi></mrow><mo>.</mo></math></span>
</li>
<li>Take care to distinguish negative coefficients from negative exponents.</li>
<li>Scientific notation is particularly useful when working with numbers that are very large or very small.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch01_s05_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd01">
<h3 class="title">Part A: Rules of Exponents</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch01_s05_qs01_p01"><strong class="emphasis bold">Simplify. (Assume all variables represent nonzero numbers.)</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1142" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mn>7</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1144" display="inline"><mrow><msup><mn>7</mn><mn>3</mn></msup><mo>⋅</mo><msup><mn>7</mn><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa03">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1146" display="block"><mrow><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mn>2</mn></msup><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>5</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa04">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1148" display="block"><mrow><mfrac><mrow><msup><mn>7</mn><mn>5</mn></msup><mo>⋅</mo><msup><mn>7</mn><mn>9</mn></msup></mrow><mrow><msup><mn>7</mn><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1150" display="inline"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1152" display="inline"><mrow><msup><mi>y</mi><mn>5</mn></msup><mo>⋅</mo><msup><mi>y</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa07">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1154" display="block"><mrow><mfrac><mrow><msup><mi>a</mi><mn>8</mn></msup><mo>⋅</mo><msup><mi>a</mi><mn>6</mn></msup></mrow><mrow><msup><mi>a</mi><mn>5</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa08">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1156" display="block"><mrow><mfrac><mrow><msup><mi>b</mi><mn>4</mn></msup><mo>⋅</mo><msup><mi>b</mi><mrow><mn>10</mn></mrow></msup></mrow><mrow><msup><mi>b</mi><mn>8</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa09">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1158" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>⋅</mo><msup><mi>x</mi><mrow><mn>3</mn><mi>n</mi></mrow></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa10">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1160" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>⋅</mo><msup><mi>x</mi><mrow><mn>8</mn><mi>n</mi></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mn>3</mn><mi>n</mi></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1162" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>5</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1164" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1166" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>4</mn></msup><msup><mi>y</mi><mn>5</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1168" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>7</mn></msup><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1170" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup><msup><mi>z</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1172" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><msup><mi>z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1174" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><msup><mi>z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1176" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mi>x</mi><msup><mi>y</mi><mn>3</mn></msup><msup><mi>z</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1178" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><msup><mi>z</mi><mn>5</mn></msup></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1180" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><msup><mi>z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1182" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>⋅</mo><msup><mi>x</mi><mn>3</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1184" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>y</mi><mn>5</mn></msup><mo>⋅</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa23">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1186" display="block"><mrow><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>⋅</mo><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa24">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1188" display="block"><mrow><mfrac><mrow><mi>a</mi><mo>⋅</mo><msup><mi>a</mi><mn>3</mn></msup><mo>⋅</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1189" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>9</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1191" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>7</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1193" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1195" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>7</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1197" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>⋅</mo><mn>3</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1199" display="inline"><mrow><mo>−</mo><mn>10</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>⋅</mo><mn>2</mn><mi>x</mi><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1201" display="inline"><mrow><mo>−</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><msup><mi>z</mi><mn>3</mn></msup><mo>⋅</mo><mn>3</mn><mi>x</mi><mi>y</mi><msup><mi>z</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p64"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1203" display="inline"><mrow><mn>2</mn><mi>x</mi><mi>y</mi><msup><mi>z</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mi>z</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1205" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mi>n</mi></msup><msup><mi>y</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>⋅</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p68"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1207" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mrow><mn>5</mn><mi>n</mi></mrow></msup><msup><mi>y</mi><mi>n</mi></msup><mo>⋅</mo><mn>2</mn><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa35">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1209" display="block"><mrow><mfrac><mrow><mn>40</mn><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mi>z</mi></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mi>z</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa36">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1211" display="block"><mrow><mfrac><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>5</mn></msup><msup><mi>z</mi><mn>3</mn></msup></mrow><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mi>z</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa37">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1213" display="block"><mrow><mfrac><mrow><mn>24</mn><msup><mi>a</mi><mn>8</mn></msup><msup><mi>b</mi><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>5</mn><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>10</mn></mrow></msup></mrow><mrow><mn>8</mn><msup><mi>a</mi><mn>5</mn></msup><msup><mi>b</mi><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>5</mn><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa38">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1215" display="block"><mrow><mfrac><mrow><mn>175</mn><msup><mi>m</mi><mn>9</mn></msup><msup><mi>n</mi><mn>5</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow><mo>)</mo></mrow></mrow><mn>7</mn></msup></mrow><mrow><mn>25</mn><msup><mi>m</mi><mn>8</mn></msup><mi>n</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa39">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1217" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mn>6</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1219" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mi>x</mi><msup><mi>y</mi><mn>4</mn></msup><msup><mi>z</mi><mn>7</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa41">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1221" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>3</mn><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>c</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa42">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1223" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>10</mn><msup><mi>a</mi><mn>3</mn></msup><mi>b</mi></mrow><mrow><mn>3</mn><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa43">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1225" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi><msup><mi>y</mi><mn>4</mn></msup></mrow><mrow><msup><mi>z</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa44">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1227" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>7</mn><msup><mi>x</mi><mn>9</mn></msup><mi>y</mi></mrow><mrow><msup><mi>z</mi><mn>4</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa45">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1229" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><msup><mi>z</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa46">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1231" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mrow><mi>z</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1233" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>0</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1234" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>0</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p98"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1235" display="inline"><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>0</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1236" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>0</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1238" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>0</mn></msup><msup><mi>c</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1240" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><msup><mi>a</mi><mn>4</mn></msup><msup><mi>b</mi><mn>2</mn></msup><msup><mi>c</mi><mn>0</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa53">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1242" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>2</mn></msup><msup><mi>z</mi><mn>0</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>3</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa54">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1244" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>0</mn></msup><msup><mi>y</mi><mn>5</mn></msup><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mrow><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup><msup><mi>z</mi><mn>0</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1246" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1248" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1250" display="inline"><mrow><msup><mi>a</mi><mn>4</mn></msup><mo>⋅</mo><msup><mi>a</mi><mrow><mo>−</mo><mn>5</mn></mrow></msup><mo>⋅</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1252" display="inline"><mrow><msup><mi>b</mi><mrow><mo>−</mo><mn>8</mn></mrow></msup><mo>⋅</mo><msup><mi>b</mi><mn>3</mn></msup><mo>⋅</mo><msup><mi>b</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa59">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1254" display="block"><mrow><mfrac><mrow><msup><mi>a</mi><mn>8</mn></msup><mo>⋅</mo><msup><mi>a</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mi>a</mi><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa60">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1256" display="block"><mrow><mfrac><mrow><msup><mi>b</mi><mrow><mo>−</mo><mn>10</mn></mrow></msup><mo>⋅</mo><msup><mi>b</mi><mn>4</mn></msup></mrow><mrow><msup><mi>b</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1258" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1260" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>5</mn></mrow></msup><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1262" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1264" display="inline"><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><msup><mi>z</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa65">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1266" display="block"><mrow><mfrac><mrow><mn>25</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa66">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1268" display="block"><mrow><mfrac><mrow><mo>−</mo><mn>9</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>y</mi><mn>3</mn></msup><msup><mi>z</mi><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><mrow><mn>3</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa67">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1270" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa68">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1272" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mrow><mo>−</mo><mn>5</mn></mrow></msup><msup><mi>z</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa69">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1274" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><mi>z</mi></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa70">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1276" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>5</mn></msup><msup><mi>z</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa71">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1278" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>12</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mi>z</mi></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>7</mn></msup><mi>y</mi><msup><mi>z</mi><mn>8</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa72">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1280" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>150</mn><mi>x</mi><msup><mi>y</mi><mn>8</mn></msup><msup><mi>z</mi><mn>2</mn></msup></mrow><mrow><mn>90</mn><msup><mi>x</mi><mn>7</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mi>z</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa73">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1282" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>9</mn><msup><mi>a</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><msup><mi>b</mi><mn>4</mn></msup><msup><mi>c</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn><msup><mi>a</mi><mn>3</mn></msup><msup><mi>b</mi><mn>5</mn></msup><msup><mi>c</mi><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa74">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1284" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>15</mn><msup><mi>a</mi><mn>7</mn></msup><msup><mi>b</mi><mn>5</mn></msup><msup><mi>c</mi><mrow><mo>−</mo><mn>8</mn></mrow></msup></mrow><mrow><mn>3</mn><msup><mi>a</mi><mrow><mo>−</mo><mn>6</mn></mrow></msup><msup><mi>b</mi><mn>2</mn></msup><msup><mi>c</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd01_qd02" start="75">
<p class="para" id="fwk-redden-ch01_s05_qs01_p150"><strong class="emphasis bold">The value in dollars of a new mobile phone can be estimated by using the formula</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1286" display="inline"><mrow><mi>V</mi><mo>=</mo><mn>210</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span><strong class="emphasis bold">, where <em class="emphasis">t</em> is the number of years after purchase.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p151">How much was the phone worth new?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p153">How much will the phone be worth in 1 year?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p155">How much will the phone be worth in 3 years?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p157">How much will the phone be worth in 10 years?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p159">How much will the phone be worth in 100 years?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p161">According to the formula, will the phone ever be worthless? Explain.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p162">The height of a particular right circular cone is equal to the square of the radius of the base, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1287" display="inline"><mrow><mi>h</mi><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span> Find a formula for the volume in terms of <em class="emphasis">r</em>.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p164">A sphere has a radius <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1289" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span> Find the volume in terms of <em class="emphasis">x</em>.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd02">
<h3 class="title">Part B: Scientific Notation</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd02_qd01" start="83">
<p class="para" id="fwk-redden-ch01_s05_qs01_p166"><strong class="emphasis bold">Convert to a decimal number.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p167"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1291" display="inline"><mrow><mn>5.2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p169"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1292" display="inline"><mrow><mn>6.02</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p171"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1293" display="inline"><mrow><mn>1.02</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p173"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1294" display="inline"><mrow><mn>7.44</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd02_qd02" start="87">
<p class="para" id="fwk-redden-ch01_s05_qs01_p175"><strong class="emphasis bold">Rewrite using scientific notation.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p176">7,050,000</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p178">430,000,000,000</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p180">0.00005001</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p182">0.000000231</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd02_qd03" start="91">
<p class="para" id="fwk-redden-ch01_s05_qs01_p184"><strong class="emphasis bold">Perform the operations.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p185"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1299" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1.2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>5</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p187"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1301" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4.8</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>20</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p189"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1303" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>9.1</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>23</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>10</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p191"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1305" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5.5</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>25</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa95">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1307" display="block"><mrow><mfrac><mrow><mn>9.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></mrow><mrow><mn>1.2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa96">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1309" display="block"><mrow><mfrac><mrow><mn>4.8</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>14</mn></mrow></msup></mrow><mrow><mn>2.4</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa97">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1311" display="block"><mrow><mfrac><mrow><mn>4</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>8</mn></mrow></msup></mrow><mrow><mn>8</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>10</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa98">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1313" display="block"><mrow><mfrac><mrow><mn>2.3</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>23</mn></mrow></msup></mrow><mrow><mn>9.2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p201"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1315" display="inline"><mrow><mn>987,000,000,000,000</mn><mo>×</mo><mn>23</mn><mo>,</mo><mn>000</mn><mo>,</mo><mn>000</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p203"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1317" display="inline"><mrow><mn>0.00000000024</mn><mo>×</mo><mn>0.00000004</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p205"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1319" display="inline"><mrow><mn>0.000000000522</mn><mo>÷</mo><mn>0.0000009</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p207"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1321" display="inline"><mrow><mn>81,000,000,000</mn><mo>÷</mo><mn>0.0000648</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p209">The population density of Earth refers to the number of people per square mile of land area. If the total land area on Earth is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1323" display="inline"><mrow><mn>5.751</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>7</mn></msup></mrow></math></span> square miles and the population in 2007 was estimated to be <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1324" display="inline"><mrow><mn>6.67</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow></math></span> people, then calculate the population density of Earth at that time.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p211">In 2008 the population of New York City was estimated to be 8.364 million people. The total land area is 305 square miles. Calculate the population density of New York City.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p213">The mass of Earth is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1325" display="inline"><mrow><mn>5.97</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></mrow></math></span> kilograms and the mass of the Moon is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1326" display="inline"><mrow><mn>7.35</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>22</mn></mrow></msup></mrow></math></span> kilograms. By what factor is the mass of Earth greater than the mass of the Moon?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p215">The mass of the Sun is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1327" display="inline"><mrow><mn>1.99</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>30</mn></mrow></msup></mrow></math></span> kilograms and the mass of Earth is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1328" display="inline"><mrow><mn>5.97</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></mrow></math></span> kilograms. By what factor is the mass of the Sun greater than the mass of Earth? Express your answer in scientific notation.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p217">The radius of the Sun is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1330" display="inline"><mrow><mn>4.322</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>5</mn></msup></mrow></math></span> miles and the average distance from Earth to the Moon is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1331" display="inline"><mrow><mn>2.392</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>5</mn></msup></mrow></math></span> miles. By what factor is the radius of the Sun larger than the average distance from Earth to the Moon?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p219">One light year, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1332" display="inline"><mrow><mn>9.461</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>15</mn></mrow></msup></mrow></math></span> meters, is the distance that light travels in a vacuum in one year. If the distance from our Sun to the nearest star, Proxima Centauri, is estimated to be <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1333" display="inline"><mrow><mn>3.991</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></mrow></math></span> meters, then calculate the number of years it would take light to travel that distance.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p221">It is estimated that there are about 1 million ants per person on the planet. If the world population was estimated to be 6.67 billion people in 2007, then estimate the world ant population at that time.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p223">The radius of the earth is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1335" display="inline"><mrow><mn>6.3</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>6</mn></msup></mrow></math></span> meters and the radius of the sun is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1336" display="inline"><mrow><mn>7.0</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow></math></span> meters. By what factor is the radius of the Sun larger than the radius of the Earth?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p225">A gigabyte is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1337" display="inline"><mrow><mn>1</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow></math></span> bytes and a megabyte is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1338" display="inline"><mrow><mn>1</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>6</mn></msup></mrow></math></span> bytes. If the average song in the MP3 format consumes about 4.5 megabytes of storage, then how many songs will fit on a 4-gigabyte memory card?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p227">Water weighs approximately 18 grams per mole. If one mole is about <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1339" display="inline"><mrow><mn>6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>23</mn></mrow></msup></mrow></math></span> molecules, then approximate the weight of each molecule of water.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s05_qs01_qd03_qd01" start="113">
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa113">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p229">Use numbers to show that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1341" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mo>≠</mo><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><msup><mi>y</mi><mi>n</mi></msup></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa114">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p230">Why is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1342" display="inline"><mrow><msup><mn>0</mn><mn>0</mn></msup></mrow></math></span> indeterminate?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa115">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p231">Explain to a beginning algebra student why <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1343" display="inline"><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><msup><mn>2</mn><mn>3</mn></msup><mo>≠</mo><msup><mn>4</mn><mn>5</mn></msup></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa116">
<div class="question">
<p class="para" id="fwk-redden-ch01_s05_qs01_p232">René Descartes (1637) established the usage of exponential form: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1344" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1345" display="inline"><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow></math></span>, and so on. Before this, how were exponents denoted?</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch01_s05_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s05_qs01_p03_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1143" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>11</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s05_qs01_p07_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1147" display="inline"><mrow><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s05_qs01_p11_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1151" display="inline"><mrow><msup><mi>x</mi><mn>5</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s05_qs01_p15_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1155" display="inline"><mrow><msup><mi>a</mi><mn>9</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s05_qs01_p19_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1159" display="inline"><mrow><msup><mi>x</mi><mrow><mn>4</mn><mi>n</mi></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s05_qs01_p23_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1163" display="inline"><mrow><msup><mi>x</mi><mrow><mn>15</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s05_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>