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<title>Roots and Radicals</title>
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<div class="section" id="fwk-redden-ch05_s01" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">5.1</span> Roots and Radicals</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch05_s01_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch05_s01_o01" numeration="arabic">
<li>Identify and evaluate square and cube roots.</li>
<li>Determine the domain of functions involving square and cube roots.</li>
<li>Evaluate <em class="emphasis">n</em>th roots.</li>
<li>Simplify radicals using the product and quotient rules for radicals.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch05_s01_s01" version="5.0" lang="en">
<h2 class="title editable block">Square and Cube Roots</h2>
<p class="para block" id="fwk-redden-ch05_s01_s01_p01">Recall that a <span class="margin_term"><a class="glossterm">square root</a><span class="glossdef">A number that when multiplied by itself yields the original number.</span></span> of a number is a number that when multiplied by itself yields the original number. For example, 5 is a square root of 25, because <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0001" display="inline"><mrow><msup><mn>5</mn><mn>2</mn></msup><mo>=</mo><mn>25</mn></mrow><mo>.</mo></math></span> Since <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0002" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>25</mn></mrow></math></span>, we can say that −5 is a square root of 25 as well. Every positive real number has two square roots, one positive and one negative. For this reason, we use the radical sign <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0003" display="inline"><mrow><msqrt><mrow><mtext> </mtext></mrow></msqrt></mrow></math></span> to denote the <span class="margin_term"><a class="glossterm">principal (nonnegative) square root</a><span class="glossdef">The positive square root of a positive real number, denoted with the symbol <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0004" display="inline"><mrow><msqrt><mrow><mtext> </mtext></mrow></msqrt></mrow><mo>.</mo></math></span></span></span> and a negative sign in front of the radical <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0005" display="inline"><mrow><mo>−</mo><msqrt><mrow><mtext> </mtext></mrow></msqrt></mrow></math></span> to denote the negative square root.</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0006" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>25</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>P</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mn>25</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><msqrt><mrow><mn>25</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>N</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mn>25</mn></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p03">Zero is the only real number with one square root.</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0007" display="block"><mrow><msqrt><mn>0</mn></msqrt><mo>=</mo><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>because</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msup><mn>0</mn><mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow></math>
</span></p>
<div class="callout block" id="fwk-redden-ch05_s01_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch05_s01_s01_p05">Evaluate.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s01_o01" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0008" display="inline"><mrow><msqrt><mrow><mn>121</mn></mrow></msqrt></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0009" display="inline"><mrow><mo>−</mo><msqrt><mrow><mn>81</mn></mrow></msqrt></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s01_o02" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0010" display="inline"><mrow><mtext> </mtext><msqrt><mrow><mn>121</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><msup><mrow><mn>11</mn></mrow><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mn>11</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0011" display="inline"><mrow><mtext> </mtext><mo>−</mo><msqrt><mrow><mn>81</mn></mrow></msqrt><mo>=</mo><mo>−</mo><msqrt><mrow><msup><mn>9</mn><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mtext>−</mtext><mn>9</mn></mrow></math></span></li>
</ol>
</div>
<p class="para block" id="fwk-redden-ch05_s01_s01_p06">If the <span class="margin_term"><a class="glossterm">radicand</a><span class="glossdef">The expression <em class="emphasis">A</em> within a radical sign, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0012" display="inline"><mrow><mroot><mi>A</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow><mo>.</mo></math></span></span></span>, the number inside the radical sign, can be factored as the square of another number, then the square root of the number is apparent. In this case, we have the following property:</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p07"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0013" display="block"><mrow><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mi>a</mi><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p08">Or more generally,</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p09"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0014" display="block"><mrow><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>a</mi><mo>∈</mo><mi>ℝ</mi></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p10">The absolute value is important because <em class="emphasis">a</em> may be a negative number and the radical sign denotes the principal square root. For example,</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0015" display="block"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mrow><mo>|</mo><mrow><mtext>−</mtext><mn>8</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>8</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p12">Make use of the absolute value to ensure a positive result.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch05_s01_s01_p13">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0016" display="inline"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p14">Here the variable expression <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0017" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span> could be negative, zero, or positive. Since the sign depends on the unknown quantity <em class="emphasis">x</em>, we must ensure that we obtain the principal square root by making use of the absolute value.</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p15"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0018" display="block"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s01_p16">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0019" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch05_s01_s01_p17">The importance of the use of the absolute value in the previous example is apparent when we evaluate using values that make the radicand negative. For example, when <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0020" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span>,</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p18"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0021" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p19">Next, consider the square root of a negative number. To determine the square root of −25, you must find a number that when squared results in −25:</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0022" display="block"><mrow><msqrt><mrow><mtext>−</mtext><mn>25</mn></mrow></msqrt><mo>=</mo><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> or </mtext><msup><mrow><mrow><mo>(</mo><mrow><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mtext> </mtext><mo>−</mo><mn>25</mn></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p21">However, any real number squared always results in a positive number. The square root of a negative number is currently left undefined. For now, we will state that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0023" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>25</mn></mrow></msqrt></mrow></math></span> is not a real number. Therefore, the <span class="margin_term"><a class="glossterm">square root function</a><span class="glossdef">The function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0024" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mi>x</mi></msqrt></mrow><mo>.</mo></math></span></span></span> given by <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0025" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mi>x</mi></msqrt></mrow></math></span> is not defined to be a real number if the <em class="emphasis">x</em>-values are negative. The smallest value in the domain is zero. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0026" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msqrt><mn>0</mn></msqrt><mo>=</mo><mn>0</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0027" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>=</mo><msqrt><mn>4</mn></msqrt><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span> Recall the graph of the square root function.</p>
<div class="informalfigure large block">
<img src="section_08/5a5bec559b23c37c3dc80885e5f0026c.png">
</div>
<p class="para block" id="fwk-redden-ch05_s01_s01_p23">The domain and range both consist of real numbers greater than or equal to zero: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0028" display="inline"><mrow><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> To determine the domain of a function involving a square root we look at the radicand and find the values that produce nonnegative results.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch05_s01_s01_p24">Determine the domain of the function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0029" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p25">Here the radicand is <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0030" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span> This expression must be zero or positive. In other words,</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p26"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0031" display="block"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>≥</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s01_p27">Solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p28"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0032" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>≥</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>≥</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>3</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>≥</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s01_p29">Answer: Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0033" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch05_s01_s01_p30">A <span class="margin_term"><a class="glossterm">cube root</a><span class="glossdef">A number that when used as a factor with itself three times yields the original number, denoted with the symbol <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0034" display="inline"><mrow><mroot><mrow><mtext> </mtext></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></span></span> of a number is a number that when multiplied by itself three times yields the original number. Furthermore, we denote a cube root using the symbol <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0035" display="inline"><mrow><mroot><mrow><mtext> </mtext></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>, where 3 is called the <span class="margin_term"><a class="glossterm">index</a><span class="glossdef">The positive integer <em class="emphasis">n</em> in the notation <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0036" display="inline"><mroot><mrow><mtext> </mtext></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></math></span> that is used to indicate an <em class="emphasis">n</em>th root.</span></span>. For example,</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0037" display="block"><mrow><mroot><mrow><mn>64</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>4</mn><mtext>, because</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msup><mn>4</mn><mn>3</mn></msup><mo>=</mo><mn>64</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p32">The product of three equal factors will be positive if the factor is positive and negative if the factor is negative. For this reason, any real number will have only one real cube root. Hence the technicalities associated with the principal root do not apply. For example,</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p33"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0038" display="block"><mrow><mroot><mrow><mtext>−</mtext><mn>64</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>4</mn><mtext>, because</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mo>=</mo><mtext>−</mtext><mn>64</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p34">In general, given any real number <em class="emphasis">a</em>, we have the following property:</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p35"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0039" display="block"><mrow><mroot><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mi>a</mi><mtext> if a</mtext><mo>∈</mo><mi>ℝ</mi></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p36">When simplifying cube roots, look for factors that are perfect cubes.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch05_s01_s01_p37">Evaluate.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s01_o03" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0040" display="inline"><mrow><mroot><mn>8</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0041" display="inline"><mrow><mroot><mn>0</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0042" display="inline"><mrow><mroot><mrow><mfrac><mn>1</mn><mrow><mn>27</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0043" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0044" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>125</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s01_o04" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0045" display="inline"><mrow><mroot><mn>8</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>2</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0046" display="inline"><mrow><mroot><mn>0</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mn>0</mn><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>0</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0047" display="inline"><mrow><mroot><mrow><mfrac><mn>1</mn><mrow><mn>27</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0048" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>1</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0049" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>125</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>5</mn></mrow></math></span></li>
</ol>
</div>
<p class="para block" id="fwk-redden-ch05_s01_s01_p38">It may be the case that the radicand is not a perfect square or cube. If an integer is not a perfect power of the index, then its root will be irrational. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0050" display="inline"><mrow><mroot><mn>2</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span> is an irrational number that can be approximated on most calculators using the root button <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0051" display="inline"><mrow><mtable columnspacing="0.1em" frame="solid"><mrow><mroot><mrow><mtext> </mtext></mrow><mpadded width="0.4em"><mi>x</mi></mpadded></mroot></mrow></mtable></mrow><mo>.</mo></math></span> Depending on the calculator, we typically type in the index prior to pushing the button and then the radicand as follows:</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p39"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0052" display="block"><mrow><mn>3</mn><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mrow><mroot><mi>y</mi><mpadded width="0.4em"><mi>x</mi></mpadded></mroot></mrow></mtable><mtext> </mtext><mn>2</mn><mtext> </mtext><mtable columnspacing="0.1em" frame="solid"><mo>=</mo></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p40">Therefore, we have</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p41"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0053" display="block"><mrow><mroot><mn>2</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>≈</mo><mn>1.260</mn><mo>,</mo><mtext> because </mtext><mn>1.260</mn><mo>^</mo><mn>3</mn><mo>≈</mo><mn>2</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p42">Since cube roots can be negative, zero, or positive we do not make use of any absolute values.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch05_s01_s01_p43">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0054" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p44">The cube root of a quantity cubed is that quantity.</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p45"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0055" display="block"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mi>y</mi><mo>−</mo><mn>7</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s01_p46">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0056" display="inline"><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s01_s01_n05a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s01_s01_p47"><strong class="emphasis bold">Try this!</strong> Evaluate: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0057" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>1000</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s01_s01_p48">Answer: −10</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/B06NIs-3gig" condition="http://img.youtube.com/vi/B06NIs-3gig/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/B06NIs-3gig" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para block" id="fwk-redden-ch05_s01_s01_p50">Next, consider the <span class="margin_term"><a class="glossterm">cube root function</a><span class="glossdef">The function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0058" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></span></span>:</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p51"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0059" display="block"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mstyle color="#007fbf"><mtext> </mtext><mi>C</mi><mi>u</mi><mi>b</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>f</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext>.</mtext></mstyle></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p52">Since the cube root could be either negative or positive, we conclude that the domain consists of all real numbers. Sketch the graph by plotting points. Choose some positive and negative values for <em class="emphasis">x</em>, as well as zero, and then calculate the corresponding <em class="emphasis">y</em>-values.</p>
<p class="para block" id="fwk-redden-ch05_s01_s01_p53"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0060" display="block"><mrow><mtable columnspacing="0.1em" columnalign="center" columnlines="solid none" rowlines="solid none"><mtr columnalign="center"><mtd columnalign="center" style="border-bottom:1pt solid black"><mi>x</mi></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>O</mi><mi>r</mi><mi>d</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>d</mi><mtext> </mtext><mi>P</mi><mi>a</mi><mi>i</mi><mi>r</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mrow><mtext>−</mtext><mn>8</mn></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mrow><mtext>−</mtext><mn>2</mn></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>8</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mtext>−</mtext><mn>8</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>2</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>8</mn><mo>,</mo><mtext>−</mtext><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mrow><mtext>−</mtext><mn>1</mn></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mrow><mtext>−</mtext><mn>1</mn></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mtext>−</mtext><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo>,</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mn>0</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mn>0</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mn>1</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mn>1</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>8</mn></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mn>8</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mn>8</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p54">Plot the points and sketch the graph of the cube root function.</p>
<div class="informalfigure large block">
<img src="section_08/507f092c47524abca6b1b6d100809a5e.png">
</div>
<p class="para editable block" id="fwk-redden-ch05_s01_s01_p56">The graph passes the vertical line test and is indeed a function. In addition, the range consists of all real numbers.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s01_n06">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch05_s01_s01_p57">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0061" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0062" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>9</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0063" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0064" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0065" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0066" display="inline"><mi>g</mi><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p58">Replace <em class="emphasis">x</em> with the given values.</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p59"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0067" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left" columnlines="solid none" rowlines="solid none"><mtr columnalign="left"><mtd columnalign="center" style="border-bottom:1pt solid black"><mi>x</mi></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>O</mi><mi>r</mi><mi>d</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>d</mi><mtext> </mtext><mi>P</mi><mi>a</mi><mi>i</mi><mi>r</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="center"><mrow><mtext>−</mtext><mn>9</mn></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mn>0</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mi>g</mi><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mtext>−</mtext><mn>9</mn></mrow></mstyle><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mstyle color="#007f3f"><mrow><mtext>−</mtext><mn>9</mn></mrow></mstyle><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mtext>−</mtext><mn>8</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>2</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>9</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="center"><mrow><mtext>−</mtext><mn>2</mn></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mn>1</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mi>g</mi><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mtext>−</mtext><mn>2</mn></mrow></mstyle><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mstyle color="#007f3f"><mrow><mtext>−</mtext><mn>2</mn></mrow></mstyle><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mtext>−</mtext><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>1</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="center"><mrow><mtext>−</mtext><mn>1</mn></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mi>g</mi><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mtext>−</mtext><mn>1</mn></mrow></mstyle><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mstyle color="#007f3f"><mrow><mtext>−</mtext><mn>1</mn></mrow></mstyle><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mn>0</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mn>3</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mi>g</mi><mo stretchy="false">(</mo><mstyle color="#007f3f"><mn>0</mn></mstyle><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mn>1</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s01_p60">We can also sketch the graph using the following translations:</p>
<p class="para" id="fwk-redden-ch05_s01_s01_p61"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0068" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mi>B</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>c</mi><mi>u</mi><mi>b</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>f</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mi>H</mi><mi>o</mi><mi>r</mi><mi>i</mi><mi>z</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mn>1</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>u</mi><mi>p</mi><mtext> </mtext><mn>2</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s01_s01_p62">Answer:</p>
<div class="informalfigure large">
<img src="section_08/c7808cb98ec1b47125d20c368e032309.png">
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch05_s01_s02" version="5.0" lang="en">
<h2 class="title editable block">nth Roots</h2>
<p class="para block" id="fwk-redden-ch05_s01_s02_p01">For any integer <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0069" display="inline"><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, we define an <span class="margin_term"><a class="glossterm"><em class="emphasis">n</em>th root</a><span class="glossdef">A number that when raised to the <em class="emphasis">n</em>th power <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0070" display="inline"><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></math></span> yields the original number.</span></span> of a positive real number as a number that when raised to the <em class="emphasis">n</em>th power yields the original number. Given any nonnegative real number <em class="emphasis">a</em>, we have the following property:</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0071" display="block"><mrow><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot><mo>=</mo><mi>a</mi><mo>,</mo><mtext> if </mtext><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p03">Here <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0072" display="inline"><mi>n</mi></math></span> is called the index and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0073" display="inline"><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow></math></span> is called the radicand. Furthermore, we can refer to the entire expression <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0074" display="inline"><mrow><mroot><mi>A</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></math></span> as a <span class="margin_term"><a class="glossterm">radical</a><span class="glossdef">Used when referring to an expression of the form <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0075" display="inline"><mrow><mroot><mi>A</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow><mo>.</mo></math></span></span></span>. When the index is an integer greater than or equal to 4, we say “fourth root,” “fifth root,” and so on. The <em class="emphasis">n</em>th root of any number is apparent if we can write the radicand with an exponent equal to the index.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s02_n01">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch05_s01_s02_p04">Simplify.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s02_o01" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0076" display="inline"><mrow><mroot><mrow><mn>81</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0077" display="inline"><mrow><mroot><mrow><mn>32</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0078" display="inline"><mrow><mroot><mn>1</mn><mpadded width="0.4em"><mn>7</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0079" display="inline"><mrow><mroot><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s02_o02" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0080" display="inline"><mrow><mroot><mrow><mn>81</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mn>3</mn><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mn>3</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0081" display="inline"><mrow><mroot><mrow><mn>32</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mn>2</mn><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mn>2</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0082" display="inline"><mrow><mroot><mn>1</mn><mpadded width="0.4em"><mn>7</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mn>1</mn><mn>7</mn></msup></mrow><mpadded width="0.4em"><mn>7</mn></mpadded></mroot><mo>=</mo><mn>1</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0083" display="inline"><mrow><mroot><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></li>
</ol>
</div>
<p class="para block" id="fwk-redden-ch05_s01_s02_p05"><em class="emphasis bolditalic">Note</em>: If the index is <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0084" display="inline"><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></math></span>, then the radical indicates a square root and it is customary to write the radical without the index; <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0085" display="inline"><mrow><mroot><mi>a</mi><mpadded width="0.4em"><mn>2</mn></mpadded></mroot><mo>=</mo><msqrt><mi>a</mi></msqrt></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p06">We have already taken care to define the principal square root of a real number. At this point, we extend this idea to <em class="emphasis">n</em>th roots when <em class="emphasis">n</em> is even. For example, 3 is a fourth root of 81, because <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0086" display="inline"><mrow><msup><mn>3</mn><mn>4</mn></msup><mo>=</mo><mn>81</mn></mrow><mo>.</mo></math></span> And since <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0087" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup><mo>=</mo><mn>81</mn></mrow></math></span>, we can say that −3 is a fourth root of 81 as well. Hence we use the radical sign <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0088" display="inline"><mrow><mroot><mrow><mtext> </mtext></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></math></span> to denote the <span class="margin_term"><a class="glossterm">principal (nonnegative) <em class="emphasis">n</em>th root</a><span class="glossdef">The positive <em class="emphasis">n</em>th root when <em class="emphasis">n</em> is even.</span></span> when <em class="emphasis">n</em> is even. In this case, for any real number <em class="emphasis">a</em>, we use the following property:</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p07"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0089" display="block"><mrow><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>=</mo><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mtext> </mtext><mstyle color="#007fbf"><mtext> </mtext><mi>W</mi><mi>h</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>n</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi></mstyle></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p08">For example,</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p09"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0090" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mroot><mrow><mn>81</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><msup><mn>3</mn><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mn>3</mn><mo>|</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mtext> </mtext></mtd></mtr><mtr><mtd columnalign="right"><mroot><mrow><mn>81</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mtext> </mtext></mtd></mtr></mtable></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p10">The negative <em class="emphasis">n</em>th root, when <em class="emphasis">n</em> is even, will be denoted using a negative sign in front of the radical <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0091" display="inline"><mrow><mo>−</mo><mroot><mrow><mtext> </mtext></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p11"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0092" display="block"><mrow><mo>−</mo><mroot><mrow><mn>81</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mo>−</mo><mroot><mrow><msup><mn>3</mn><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>3</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p12">We have seen that the square root of a negative number is not real because any real number that is squared will result in a positive number. In fact, a similar problem arises for any even index:</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p13"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0093" display="block"><mrow><mroot><mrow><mtext>−</mtext><mn>81</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><msup><mrow><mrow><mo>(</mo><mrow><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup><mo>=</mo><mtext>−</mtext><mn>81</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p14">We can see that a fourth root of −81 is not a real number because the fourth power of any real number is always positive.</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0094" display="block"><mrow><mrow><mtable columnspacing="0.1em" columnalign="right"><mtr><mtd><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt></mtd></mtr><mtr><mtd><mroot><mrow><mtext>−</mtext><mn>81</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mtd></mtr><mtr><mtd><mroot><mrow><mtext>−</mtext><mn>64</mn></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mtd></mtr></mtable><mo>}</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>T</mi><mi>h</mi><mi>e</mi><mi>s</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>s</mi><mtext> </mtext><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>e</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>n</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>s</mi><mtext>.</mtext></mstyle></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p16">You are encouraged to try all of these on a calculator. What does it say?</p>
<div class="callout block" id="fwk-redden-ch05_s01_s02_n02">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch05_s01_s02_p17">Simplify.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s02_o03" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0095" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0096" display="inline"><mrow><mroot><mrow><mo>−</mo><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0097" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>6</mn></msup></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s02_p18">Since the indices are even, use absolute values to ensure nonnegative results.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s02_o04" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0098" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mrow><mo>|</mo><mrow><mtext>−</mtext><mn>10</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>10</mn></mrow></math></span></li>
<li>
<span class="inlineequation"><math xml:id="fwk-redden-ch05_m0099" display="inline"><mrow><mroot><mrow><mo>−</mo><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mtext>−</mtext><mn>10,000</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span> is not a real number.</li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0100" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>6</mn></msup></mrow><mpadded width="0.4em"><mn>6</mn></mpadded></mroot><mo>=</mo><mrow><mo>|</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow></mrow></math></span></li>
</ol>
</div>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p19">When the index <em class="emphasis">n</em> is odd, the same problems do not occur. The product of an odd number of positive factors is positive and the product of an odd number of negative factors is negative. Hence when the index <em class="emphasis">n</em> is odd, there is only one real <em class="emphasis">n</em>th root for any real number <em class="emphasis">a</em>. And we have the following property:</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p20"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0101" display="block"><mrow><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot><mo>=</mo><mi>a</mi><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>W</mi><mi>h</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>n</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>o</mi><mi>d</mi><mi>d</mi></mstyle></mrow></math>
</span></p>
<div class="callout block" id="fwk-redden-ch05_s01_s02_n03">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch05_s01_s02_p21">Simplify.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s02_o05" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0102" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0103" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>32</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0104" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>7</mn></msup></mrow><mpadded width="0.4em"><mn>7</mn></mpadded></mroot></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s02_p22">Since the indices are odd, the absolute value is not used.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s01_s02_o06" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0105" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>10</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0106" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>32</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>2</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0107" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>7</mn></msup></mrow><mpadded width="0.4em"><mn>7</mn></mpadded></mroot><mo>=</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow></math></span></li>
</ol>
</div>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p23">In summary, for any real number <em class="emphasis">a</em> we have,</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p24"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0108" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>|</mo><mtext> </mtext><mi>a</mi><mtext> </mtext><mo>|</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>W</mi><mi>h</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>n</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>W</mi><mi>h</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>n</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>o</mi><mi>d</mi><mi>d</mi></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p25">When <em class="emphasis">n is odd</em>, the <em class="emphasis">n</em>th root is <em class="emphasis">positive or negative</em> depending on the sign of the radicand.</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p26"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0109" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mroot><mrow><mn>27</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><msup><mn>3</mn><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mroot><mrow><mtext>−</mtext><mn>27</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s01_s02_p27">When <em class="emphasis">n is even</em>, the <em class="emphasis">n</em>th root is <em class="emphasis">positive or not real</em> depending on the sign of the radicand.</p>
<p class="para block" id="fwk-redden-ch05_s01_s02_p28"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0110" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>2</mn><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mo>−</mo><mn>2</mn><mo>|</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mroot><mrow><mtext>−</mtext><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>N</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mi>r</mi><mi>e</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>n</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi></mrow></mstyle></mtd></mtr></mtable></math>
</span></p>
<div class="callout block" id="fwk-redden-ch05_s01_s02_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s01_s02_p29"><strong class="emphasis bold">Try this!</strong> Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0111" display="inline"><mrow><mtext>−</mtext><mn>8</mn><mroot><mrow><mtext>−</mtext><mn>32</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s01_s02_p30">Answer: 16</p>
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</div>
<div class="section" id="fwk-redden-ch05_s01_s03" version="5.0" lang="en">
<h2 class="title editable block">Simplifying Radicals</h2>
<p class="para block" id="fwk-redden-ch05_s01_s03_p01">It will not always be the case that the radicand is a perfect power of the given index. If it is not, then we use the <span class="margin_term"><a class="glossterm">product rule for radicals</a><span class="glossdef">Given real numbers <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0112" display="inline"><mrow><mroot><mi>A</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0113" display="inline"><mrow><mroot><mi>B</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0114" display="inline"><mrow><mtext> </mtext><mroot><mrow><mi>A</mi><mo>⋅</mo><mi>B</mi></mrow><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>=</mo><mroot><mi>A</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>⋅</mo><mroot><mi>B</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow><mo>.</mo></math></span></span></span> and the <span class="margin_term"><a class="glossterm">quotient rule for radicals</a><span class="glossdef">Given real numbers <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0115" display="inline"><mrow><mroot><mi>A</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0116" display="inline"><mrow><mroot><mi>B</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0117" display="inline"><mrow><mtext> </mtext><mroot><mrow><mfrac><mi>A</mi><mi>B</mi></mfrac></mrow><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>=</mo><mfrac><mrow><mroot><mi>A</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow><mrow><mroot><mi>B</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></mfrac></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0118" display="inline"><mrow><mi>B</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span> to simplify them. Given real numbers <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0119" display="inline"><mrow><mroot><mi>A</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0120" display="inline"><mrow><mroot><mi>B</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></math></span>,</p>
<p class="para block" id="fwk-redden-ch05_s01_s03_p02">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Product Rule for Radicals:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0121" display="inline"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>A</mi><mo>⋅</mo><mi>B</mi></mrow><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mi>A</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>⋅</mo><mroot><mi>B</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Quotient Rule for Radicals:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0122" display="inline"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle mathvariant="bold"><mtext></mtext></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mroot><mrow><mfrac><mi>A</mi><mi>B</mi></mfrac></mrow><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mroot><mi>A</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow><mrow><mroot><mi>B</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch05_s01_s03_p03">A <span class="margin_term"><a class="glossterm">radical is simplified</a><span class="glossdef">A radical where the radicand does not consist of any factors that can be written as perfect powers of the index.</span></span> if it does not contain any factors that can be written as perfect powers of the index.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s03_n01">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch05_s01_s03_p04">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0123" display="inline"><mrow><msqrt><mrow><mn>150</mn></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p05">Here 150 can be written as <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0124" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><msup><mn>5</mn><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p06"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0125" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>150</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><msup><mn>5</mn><mn>2</mn></msup></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>d</mi><mi>u</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn></mrow></msqrt><mo>⋅</mo><msqrt><mrow><msup><mn>5</mn><mn>2</mn></msup></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mn>6</mn></msqrt><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msqrt><mn>6</mn></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p07">We can verify our answer on a calculator:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p08"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0126" display="block"><mrow><msqrt><mrow><mn>150</mn></mrow></msqrt><mo>≈</mo><mn>12.25</mn><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mn>5</mn><msqrt><mn>6</mn></msqrt><mo>≈</mo><mn>12.25</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p09">Also, it is worth noting that</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p10"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0127" display="block"><mrow><msup><mrow><mn>12.25</mn></mrow><mn>2</mn></msup><mo>≈</mo><mn>150</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p11">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0128" display="inline"><mrow><mn>5</mn><msqrt><mn>6</mn></msqrt></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch05_s01_s03_p12"><strong class="emphasis bold">Note</strong>: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0129" display="inline"><mrow><mn>5</mn><msqrt><mn>6</mn></msqrt></mrow></math></span> is the exact answer and 12.25 is an approximate answer. We present exact answers unless told otherwise.</p>
<div class="callout block" id="fwk-redden-ch05_s01_s03_n02">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch05_s01_s03_p13">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0130" display="inline"><mrow><mroot><mrow><mn>160</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p14">Use the prime factorization of 160 to find the largest perfect cube factor:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p15"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0131" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>160</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mn>5</mn></msup><mo>⋅</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow></mstyle><mo>⋅</mo><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p16">Replace the radicand with this factorization and then apply the product rule for radicals.</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p17"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0132" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>160</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>2</mn><mn>3</mn></msup><mo>⋅</mo><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><mn>5</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>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>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>⋅</mo><mroot><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><mn>5</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>⋅</mo><mroot><mrow><mn>20</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p18">We can verify our answer on a calculator.</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p19"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0133" display="block"><mrow><mroot><mrow><mn>160</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>≈</mo><mn>5.43</mn><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mn>2</mn><mroot><mrow><mn>20</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>≈</mo><mn>5.43</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p20">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0134" display="inline"><mrow><mn>2</mn><mroot><mrow><mn>20</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s01_s03_n03">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch05_s01_s03_p21">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0135" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>320</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p22">Here we note that the index is odd and the radicand is negative; hence the result will be negative. We can factor the radicand as follows:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p23"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0136" display="block"><mrow><mtext> </mtext><mo>−</mo><mn>320</mn><mo>=</mo><mtext>−</mtext><mn>1</mn><mo>⋅</mo><mn>32</mn><mo>⋅</mo><mn>10</mn><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup><mo>⋅</mo><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mn>5</mn></msup><mo>⋅</mo><mn>10</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p24">Then simplify:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p25"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0137" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mtext>−</mtext><mn>320</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup><mo>⋅</mo><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mn>5</mn></msup><mo>⋅</mo><mn>10</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>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>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>⋅</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mn>5</mn></msup></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>⋅</mo><mroot><mrow><mn>10</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>1</mn><mo>⋅</mo><mn>2</mn><mo>⋅</mo><mroot><mrow><mn>10</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>2</mn><mo>⋅</mo><mroot><mrow><mn>10</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p26">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0138" display="inline"><mrow><mtext>−</mtext><mn>2</mn><mroot><mrow><mn>10</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s01_s03_n04">
<h3 class="title">Example 13</h3>
<p class="para" id="fwk-redden-ch05_s01_s03_p27">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0139" display="inline"><mrow><mroot><mrow><mo>−</mo><mfrac><mn>8</mn><mrow><mn>64</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p28">In this case, consider the equivalent fraction with <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0140" display="inline"><mrow><mtext>−</mtext><mn>8</mn><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span> in the numerator and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0141" display="inline"><mrow><mn>64</mn><mo>=</mo><msup><mn>4</mn><mn>3</mn></msup></mrow></math></span> in the denominator and then simplify.</p>
<p class="para" id="fwk-redden-ch05_s01_s03_p29"><span class="informalequation"><math xml:id="fwk-redden-ch05_m0142" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mo>−</mo><mfrac><mn>8</mn><mrow><mn>64</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mtext> </mtext><mfrac><mrow><mtext>−</mtext><mn>8</mn></mrow><mrow><mn>64</mn></mrow></mfrac><mtext> </mtext></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow><mrow><mroot><mrow><msup><mn>4</mn><mn>3</mn></msup></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>−</mtext><mn>2</mn></mrow><mn>4</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p30">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0143" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s01_s03_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s01_s03_p31"><strong class="emphasis bold">Try this!</strong> Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0144" display="inline"><mrow><mroot><mrow><mfrac><mrow><mn>80</mn></mrow><mrow><mn>81</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
<p class="para" id="fwk-redden-ch05_s01_s03_p32">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0145" display="inline"><mrow><mfrac><mrow><mn>2</mn><mroot><mn>5</mn><mpadded width="0.6em"><mn>4</mn></mpadded></mroot></mrow><mn>3</mn></mfrac></mrow></math></span></p>
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</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch05_s01_s03_n05">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch05_s01_s03_l01" mark="bullet">
<li>To simplify a square root, look for the largest perfect square factor of the radicand and then apply the product or quotient rule for radicals.</li>
<li>To simplify a cube root, look for the largest perfect cube factor of the radicand and then apply the product or quotient rule for radicals.</li>
<li>When working with <em class="emphasis">n</em>th roots, <em class="emphasis">n</em> determines the definition that applies. We use <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0146" display="inline"><mrow><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot><mo>=</mo><mi>a</mi></mrow></math></span> when <em class="emphasis">n</em> is odd and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0147" display="inline"><mrow><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot><mo>=</mo><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow></mrow></math></span> when <em class="emphasis">n</em> is even.</li>
<li>To simplify <em class="emphasis">n</em>th roots, look for the factors that have a power that is equal to the index <em class="emphasis">n</em> and then apply the product or quotient rule for radicals. Typically, the process is streamlined if you work with the prime factorization of the radicand.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch05_s01_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd01">
<h3 class="title">Part A: Square and Cube Roots</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch05_s01_qs01_p01"><strong class="emphasis bold">Simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0148" display="inline"><mrow><msqrt><mrow><mn>36</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0149" display="inline"><mrow><msqrt><mrow><mn>100</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa03">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0150" display="block"><mrow><msqrt><mrow><mfrac><mn>4</mn><mn>9</mn></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa04">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0152" display="block"><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>64</mn></mrow></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0154" display="inline"><mrow><mo>−</mo><msqrt><mrow><mn>16</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0155" display="inline"><mrow><mo>−</mo><msqrt><mn>1</mn></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0156" display="inline"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0157" display="inline"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0158" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0159" display="inline"><mrow><msqrt><mrow><mo>−</mo><msup><mn>5</mn><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0160" display="inline"><mrow><mo>−</mo><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0161" display="inline"><mrow><mo>−</mo><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0162" display="inline"><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0164" display="inline"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0166" display="inline"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0168" display="inline"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0170" display="inline"><mrow><mroot><mrow><mn>64</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0171" display="inline"><mrow><mroot><mrow><mn>216</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0172" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>216</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0173" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>64</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0174" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>8</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0175" display="inline"><mrow><mroot><mn>1</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0176" display="inline"><mrow><mo>−</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0177" display="inline"><mrow><mo>−</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa25">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0178" display="block"><mrow><mroot><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa26">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0180" display="block"><mrow><mroot><mrow><mfrac><mn>8</mn><mrow><mn>27</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0182" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0184" display="inline"><mrow><mo>−</mo><mroot><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0186" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0188" display="inline"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd01_qd02" start="31">
<p class="para" id="fwk-redden-ch05_s01_qs01_p62"><strong class="emphasis bold">Determine the domain of the given function.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0190" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0192" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0194" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0196" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0198" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0200" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mo>−</mo><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0202" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>5</mn><mo>−</mo><mi>x</mi></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0204" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0206" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0208" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd01_qd03" start="41">
<p class="para" id="fwk-redden-ch05_s01_qs01_p83"><strong class="emphasis bold">Evaluate given the function definition.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p84">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0210" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0211" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0212" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0213" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p86">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0217" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0218" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>5</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0219" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0220" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>20</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p88">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0224" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>3</mn></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0225" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0226" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0227" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p90">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0231" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>5</mn></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0232" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0233" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0234" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>25</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p92">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0238" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0239" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0240" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0241" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p94">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0245" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>2</mn></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0246" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0247" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0248" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p96">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0252" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0253" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>15</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0254" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mtext>−</mtext><mn>7</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0255" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>20</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p98">Given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0259" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0260" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0261" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m0262" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd01_qd04" start="49">
<p class="para" id="fwk-redden-ch05_s01_qs01_p100"><strong class="emphasis bold">Sketch the graph of the given function and give its domain and range.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0266" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0269" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0272" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0275" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0278" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0281" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0284" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0287" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p125"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0290" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0293" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0296" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0299" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mroot><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd02">
<h3 class="title">Part B: <em class="emphasis">n</em>th Roots</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd02_qd01" start="61">
<p class="para" id="fwk-redden-ch05_s01_qs01_p137"><strong class="emphasis bold">Simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0302" display="inline"><mrow><mroot><mrow><mn>64</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p140"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0303" display="inline"><mrow><mroot><mrow><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p142"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0304" display="inline"><mrow><mroot><mrow><mn>625</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p144"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0305" display="inline"><mrow><mroot><mn>1</mn><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p146"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0306" display="inline"><mrow><mroot><mrow><mn>256</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p148"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0307" display="inline"><mrow><mroot><mrow><mn>10</mn><mo>,</mo><mn>000</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p150"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0308" display="inline"><mrow><mroot><mrow><mn>243</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p152"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0309" display="inline"><mrow><mroot><mrow><mn>100</mn><mo>,</mo><mn>000</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa69">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0310" display="block"><mrow><mroot><mrow><mfrac><mn>1</mn><mrow><mn>32</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa70">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0312" display="block"><mrow><mroot><mrow><mfrac><mn>1</mn><mrow><mn>243</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0314" display="inline"><mrow><mo>−</mo><mroot><mrow><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0315" display="inline"><mrow><mo>−</mo><mroot><mn>1</mn><mpadded width="0.4em"><mn>6</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0316" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>32</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0317" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>1</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0318" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>1</mn></mrow><mrow></mrow></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0319" display="inline"><mrow><mroot><mrow><mtext>−</mtext><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0320" display="inline"><mrow><mtext>−</mtext><mn>6</mn><mroot><mrow><mtext>−</mtext><mn>27</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0321" display="inline"><mrow><mtext>−</mtext><mn>5</mn><mroot><mrow><mtext>−</mtext><mn>8</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p174"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0322" display="inline"><mrow><mn>2</mn><mroot><mrow><mtext>−</mtext><mn>1</mn><mo>,</mo><mn>000</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0323" display="inline"><mrow><mn>7</mn><mroot><mrow><mtext>−</mtext><mn>243</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0324" display="inline"><mrow><mn>6</mn><mroot><mrow><mtext>−</mtext><mn>16</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0325" display="inline"><mrow><mn>12</mn><mroot><mrow><mtext>−</mtext><mn>64</mn></mrow><mpadded width="0.6em"><mn>6</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa83">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0326" display="block"><mrow><mn>3</mn><msqrt><mrow><mfrac><mrow><mn>25</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa84">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0328" display="block"><mrow><mn>6</mn><msqrt><mrow><mfrac><mrow><mn>16</mn></mrow><mn>9</mn></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa85">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0329" display="block"><mrow><mn>5</mn><mroot><mrow><mfrac><mrow><mn>27</mn></mrow><mrow><mn>125</mn></mrow></mfrac></mrow><mpadded width="0.5em"><mn>3</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa86">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0330" display="block"><mrow><mn>7</mn><mroot><mrow><mfrac><mrow><mn>32</mn></mrow><mrow><msup><mn>7</mn><mn>5</mn></msup></mrow></mfrac></mrow><mpadded width="0.5em"><mn>5</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa87">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0331" display="block"><mrow><mtext>−</mtext><mn>5</mn><mroot><mrow><mfrac><mn>8</mn><mrow><mn>27</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa88">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0333" display="block"><mrow><mtext>−</mtext><mn>8</mn><mroot><mrow><mfrac><mrow><mn>625</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0334" display="inline"><mrow><mn>2</mn><mroot><mrow><mn>100</mn><mo>,</mo><mn>000</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0335" display="inline"><mrow><mn>2</mn><mroot><mrow><mn>128</mn></mrow><mpadded width="0.4em"><mn>7</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd03">
<h3 class="title">Part C: Simplifying Radicals</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s01_qs01_qd03_qd01" start="91">
<p class="para" id="fwk-redden-ch05_s01_qs01_p198"><strong class="emphasis bold">Simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p199"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0336" display="inline"><mrow><msqrt><mrow><mn>96</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p201"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0338" display="inline"><mrow><msqrt><mrow><mn>500</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p203"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0340" display="inline"><mrow><msqrt><mrow><mn>480</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p205"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0342" display="inline"><mrow><msqrt><mrow><mn>450</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p207"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0344" display="inline"><mrow><msqrt><mrow><mn>320</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p209"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0346" display="inline"><mrow><msqrt><mrow><mn>216</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p211"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0348" display="inline"><mrow><mn>5</mn><msqrt><mrow><mn>112</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p213"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0350" display="inline"><mrow><mn>10</mn><msqrt><mrow><mn>135</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p215"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0352" display="inline"><mrow><mtext>−</mtext><mn>2</mn><msqrt><mrow><mn>240</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p217"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0354" display="inline"><mrow><mtext>−</mtext><mn>3</mn><msqrt><mrow><mn>162</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa101">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0356" display="block"><mrow><msqrt><mrow><mfrac><mrow><mn>150</mn></mrow><mrow><mn>49</mn></mrow></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa102">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0358" display="block"><mrow><msqrt><mrow><mfrac><mrow><mn>200</mn></mrow><mn>9</mn></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa103">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0360" display="block"><mrow><msqrt><mrow><mfrac><mrow><mn>675</mn></mrow><mrow><mn>121</mn></mrow></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa104">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0362" display="block"><mrow><msqrt><mrow><mfrac><mrow><mn>192</mn></mrow><mrow><mn>81</mn></mrow></mfrac></mrow></msqrt></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p227"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0364" display="inline"><mrow><mroot><mrow><mn>54</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p229"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0366" display="inline"><mrow><mroot><mrow><mn>24</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p231"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0368" display="inline"><mrow><mroot><mrow><mn>48</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p233"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0370" display="inline"><mrow><mroot><mrow><mn>81</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p235"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0372" display="inline"><mrow><mroot><mrow><mn>40</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p237"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0374" display="inline"><mrow><mroot><mrow><mn>120</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p239"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0376" display="inline"><mrow><mroot><mrow><mn>162</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p241"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0378" display="inline"><mrow><mroot><mrow><mn>500</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa113">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0380" display="block"><mrow><mroot><mrow><mfrac><mrow><mn>54</mn></mrow><mrow><mn>125</mn></mrow></mfrac></mrow><mpadded width="0.6em"><mn>3</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa114">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0382" display="block"><mrow><mroot><mrow><mfrac><mrow><mn>40</mn></mrow><mrow><mn>343</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa115">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p247"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0384" display="inline"><mrow><mn>5</mn><mroot><mrow><mtext>−</mtext><mn>48</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa116">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p249"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0386" display="inline"><mrow><mn>2</mn><mroot><mrow><mtext>−</mtext><mn>108</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa117">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p251"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0388" display="inline"><mrow><mn>8</mn><mroot><mrow><mn>96</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa118">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p253"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0390" display="inline"><mrow><mn>7</mn><mroot><mrow><mn>162</mn></mrow><mpadded width="0.4em"><mn>4</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa119">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p255"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0392" display="inline"><mrow><mroot><mrow><mn>160</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa120">
<div class="question">
<p class="para" id="fwk-redden-ch05_s01_qs01_p257"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m0394" display="inline"><mrow><mroot><mrow><mn>486</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa121">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0396" display="block"><mrow><mroot><mrow><mfrac><mrow><mn>224</mn></mrow><mrow><mn>243</mn></mrow></mfrac></mrow><mpadded width="0.6em"><mn>5</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa122">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m0398" display="block"><mrow><mroot><mrow><mfrac><mn>5</mn><mrow><mn>32</mn></mrow></mfrac></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s01_qs01_qa123">