<h2>Using the Product Property to Simplify Expressions With Higher Roots</h2>
<p>We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An <em>n</em>th root is considered simplified if it has no factors of \({m}^{n}\).</p>
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<h3>Simplified <em>n</em>th Root</h3>
<p>\(\sqrt[n]{a}\) is considered simplified if \(a\) has no factors of \({m}^{n}\).</p>
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<p>We will generalize the Product Property of Square Roots to include any integer root \(n\ge 2\).</p>
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<h3>Product Property of <em>n</em>th Roots</h3>
<div>\(\begin{array}{ccc}\sqrt[n]{ab}=\sqrt[n]{a}·\sqrt[n]{b}\hfill &amp; \text{and}\hfill &amp; \sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}\hfill \\ \end{array}\)</div>
<p>when \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are real numbers and for any integer \(n\ge 2\)</p>
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<h2>Example</h2>
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<div data-type="example">
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<div>
<p>Simplify:</p>
<ol>
<li>\(\sqrt[3]{{x}^{4}}\)</li>
<li>\(\sqrt[4]{{x}^{7}}\).</li>
</ol>
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<div>
<h3>Solution</h3>
<ol type="1">
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \sqrt[3]{{x}^{4}}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{largest perfect cube factor.}\hfill \end{array}\hfill &amp; &amp; \sqrt[3]{{x}^{3}·x}\hfill \\ \text{Rewrite the radical as the product of two radicals.}\hfill &amp; &amp; \sqrt[3]{{x}^{3}}·\sqrt[3]{x}\hfill \\ \text{Simplify.}\hfill &amp; &amp; x\sqrt[3]{x}\hfill \end{array}\)</li>
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \sqrt[4]{{x}^{7}}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{greatest perfect fourth power factor.}\hfill \end{array}\hfill &amp; &amp; \sqrt[4]{{x}^{4}·{x}^{3}}\hfill \\ \text{Rewrite the radical as the product of two radicals.}\hfill &amp; &amp; \sqrt[4]{{x}^{4}}·\sqrt[4]{{x}^{3}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; |x|\sqrt[4]{{x}^{3}}\hfill \end{array}\)</li>
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<h2>Example</h2>
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<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<ol>
<li>\(\sqrt[3]{16}\)</li>
<li>\(\sqrt[4]{243}\).</li>
</ol>
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<div>
<h3>Solution</h3>
<ol type="1">
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \sqrt[3]{16}\hfill \\ &amp; &amp; \sqrt[3]{{2}^{4}}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{greatest perfect cube factor.}\hfill \end{array}\hfill &amp; &amp; \sqrt[3]{{2}^{3}·2}\hfill \\ \text{Rewrite the radical as the product of two radicals.}\hfill &amp; &amp; \sqrt[3]{{2}^{3}}·\sqrt[3]{2}\hfill \\ \text{Simplify.}\hfill &amp; &amp; 2\sqrt[3]{2}\hfill \end{array}\)</li>
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \sqrt[4]{243}\hfill \\ &amp; &amp; \sqrt[4]{{3}^{5}}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{greatest perfect fourth power factor.}\hfill \end{array}\hfill &amp; &amp; \sqrt[4]{{3}^{4}·3}\hfill \\ \text{Rewrite the radical as the product of two radicals.}\hfill &amp; &amp; \sqrt[4]{{3}^{4}}·\sqrt[4]{3}\hfill \\ \text{Simplify.}\hfill &amp; &amp; 3\sqrt[4]{3}\hfill \end{array}\)</li>
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<p>Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.</p>
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<h2>Example</h2>
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<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<ol>
<li>\(\sqrt[3]{24{x}^{7}}\)</li>
<li>\(\sqrt[4]{80{y}^{14}}\).</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<ol type="1">
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \phantom{\rule{8em}{0ex}}\sqrt[3]{24{x}^{7}}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using}\hfill \\ \text{perfect cube factors.}\hfill \end{array}\hfill &amp; &amp; \phantom{\rule{8em}{0ex}}\sqrt[3]{{2}^{3}{x}^{6}·3x}\hfill \\ \text{Rewrite the radical as the product of two radicals.}\hfill &amp; &amp; \phantom{\rule{8em}{0ex}}\sqrt[3]{{2}^{3}{x}^{6}}·\sqrt[3]{3x}\hfill \\ \text{Rewrite the first radicand as}\phantom{\rule{0.2em}{0ex}}{\left(2{x}^{2}\right)}^{3}.\hfill &amp; &amp; \phantom{\rule{8em}{0ex}}\sqrt[3]{{\left(2{x}^{2}\right)}^{3}}·\sqrt[3]{3x}\hfill \\ \text{Simplify.}\hfill &amp; &amp; \phantom{\rule{8em}{0ex}}2{x}^{2}\sqrt[3]{3x}\hfill \end{array}\)</li>
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \sqrt[4]{80{y}^{14}}\hfill \\ \\ \begin{array}{c}\text{Rewrite the radicand as a product using}\phantom{\rule{0.2em}{0ex}}\text{perfect fourth power factors.}\hfill \end{array}\hfill &amp; &amp; \sqrt[4]{{2}^{4}{y}^{12}·5{y}^{2}}\hfill \\ \\ \text{Rewrite the radical as the product of two radicals.}\hfill &amp; &amp; \sqrt[4]{{2}^{4}{y}^{12}}·\sqrt[4]{5{y}^{2}}\hfill \\ \\ \text{Rewrite the first radicand as}\phantom{\rule{0.2em}{0ex}}{\left(2{y}^{3}\right)}^{4}.\hfill &amp; &amp; \sqrt[4]{{\left(2{y}^{3}\right)}^{4}}·\sqrt[4]{5{y}^{2}}\hfill \\ \\ \text{Simplify.}\hfill &amp; &amp; 2|{y}^{3}|\sqrt[4]{5{y}^{2}}\hfill \end{array}\)</li>
</ol>
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<h2>Example</h2>
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<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<ol>
<li>\(\sqrt[3]{-27}\)</li>
<li>\(\sqrt[4]{-16}\).</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<ol type="1">
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \phantom{\rule{2em}{0ex}}\sqrt[3]{-27}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using}\hfill \\ \text{perfect cube factors.}\hfill \end{array}\hfill &amp; &amp; \phantom{\rule{2em}{0ex}}\sqrt[3]{{\left(-3\right)}^{3}}\hfill \\ \text{Take the cube root.}\hfill &amp; &amp; \phantom{\rule{2em}{0ex}}-3\hfill \end{array}\)</li>
<li>
<div> </div>

\(\begin{array}{ccc}&amp; &amp; \sqrt[4]{-16}\hfill \\ \text{There is no real number}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}{n}^{4}=-16.\hfill &amp; &amp; \text{Not a real number.}\hfill \end{array}\)</li>
</ol>
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Using the Product Property to Simplify Expressions With Higher Roots

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Explore square roots and radicals, simplify expressions with them, estimate and approximate their values, perform arithmetic operations, solve radical equations, and simplify expressions with higher roots and rational exponents.


Using the Product Property to Simplify Expressions With Higher Roots

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