<h2>Simplifying Expressions with \({a}^{\frac{1}{n}}\)</h2>
<p>Rational exponents are another way of writing expressions with radicals. When we use <strong>rational exponents</strong>, we can apply the properties of exponents to simplify expressions.</p>
<p>The Power Property for Exponents says that \({\left({a}^{m}\right)}^{n}={a}^{m·n}\) when <em>m</em> and <em>n</em> are whole numbers. Let’s assume we are now not limited to whole numbers.</p>
<p>Suppose we want to find a number <em>p</em> such that \({\left({8}^{p}\right)}^{3}=8\). We will use the Power Property of Exponents to find the value of <em>p</em>.</p>
<p>\(\begin{array}{cccccc}&amp; &amp; &amp; \hfill {\left({8}^{p}\right)}^{3}&amp; =\hfill &amp; 8\hfill \\ \text{Multiply the exponents on the left.}\hfill &amp; &amp; &amp; \hfill {8}^{3p}&amp; =\hfill &amp; 8\hfill \\ \text{Write the exponent 1 on the right.}\hfill &amp; &amp; &amp; \hfill {8}^{3p}&amp; =\hfill &amp; {8}^{1}\hfill \\ \text{The exponents must be equal.}\hfill &amp; &amp; &amp; \hfill 3p&amp; =\hfill &amp; 1\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}p.\hfill &amp; &amp; &amp; \hfill p&amp; =\hfill &amp; \frac{1}{3}\hfill \\ \\ &amp; &amp; &amp; \hfill \text{So}\phantom{\rule{0.2em}{0ex}}{\left({8}^{\frac{1}{3}}\right)}^{3}&amp; =\hfill &amp; 8.\hfill \end{array}\)</p>
<p>But we know also \({\left(\sqrt[3]{8}\right)}^{3}=8\). Then it must be that \({8}^{\frac{1}{3}}=\sqrt[3]{8}\).</p>
<p>This same logic can be used for any positive integer exponent <em>n</em> to show that \({a}^{\frac{1}{n}}=\sqrt[n]{a}\).</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Rational Exponent \({a}^{\frac{1}{n}}\)</h3>
<p>If \(\sqrt[n]{a}\) is a real number and \(n\ge 2\), \({a}^{\frac{1}{n}}=\sqrt[n]{a}\).</p>
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<p>There will be times when working with expressions will be easier if you use <strong>rational exponents</strong> and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Write as a radical expression:</p>
<p>(a) \({x}^{\frac{1}{2}}\)<br/>
(b) \({y}^{\frac{1}{3}}\)<br/>
(c) \({z}^{\frac{1}{4}}\).</p>
</div>
<h3>Solution</h3>
<p>We want to write each expression in the form \(\sqrt[n]{a}\).</p>
<div>(a)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{2em}{0ex}}{x}^{\frac{1}{2}}\hfill \\ \begin{array}{c}\text{The denominator of the exponent is 2, so}\hfill \\ \text{the index of the radical is 2. We do not}\hfill \\ \text{show the index when it is 2.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{2em}{0ex}}\sqrt{x}\hfill \end{array}\)</p>
<div>(b)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{2em}{0ex}}{y}^{\frac{1}{3}}\hfill \\ \begin{array}{c}\text{The denominator of the exponent is 3, so}\hfill \\ \text{the index is 3.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{2em}{0ex}}\sqrt[3]{y}\hfill \end{array}\)</p>
<div>(c)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{2em}{0ex}}{z}^{\frac{1}{4}}\hfill \\ \begin{array}{c}\text{The denominator of the exponent is 4, so}\hfill \\ \text{the index is 4.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{2em}{0ex}}\sqrt[4]{z}\hfill \end{array}\)</p>
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<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Write with a rational exponent:</p>
<p>(a) \(\sqrt{x}\)<br/>
(b) \(\sqrt[3]{y}\)<br/>
(c) \(\sqrt[4]{z}\).</p>
</div>
<h3>Solution</h3>
<p>We want to write each radical in the form \({a}^{\frac{1}{n}}\).</p>
<div>(a)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \sqrt{x}\hfill \\ \begin{array}{c}\text{No index is shown, so it is 2.}\hfill \\ \text{The denominator of the exponent will be 2.}\hfill \end{array}\hfill &amp; &amp; &amp; {x}^{\frac{1}{2}}\hfill \end{array}\)</p>
<div>(b)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{1em}{0ex}}\sqrt[3]{y}\hfill \\ \begin{array}{c}\text{The index is 3, so the denominator of the}\hfill \\ \text{exponent is 3.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{1em}{0ex}}{y}^{\frac{1}{3}}\hfill \end{array}\)</p>
<div>(c)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{1em}{0ex}}\sqrt[4]{z}\hfill \\ \begin{array}{c}\text{The index is 4, so the denominator of the}\hfill \\ \text{exponent is 4.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{1em}{0ex}}{z}^{\frac{1}{4}}\hfill \end{array}\)</p>
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</div>
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</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Write with a rational exponent:</p>
<ol>
<li>\(\sqrt{5y}\)</li>
<li>\(\sqrt[3]{4x}\)</li>
<li>\(3\sqrt[4]{5z}\).</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<p>We want to write each radical in the form \({a}^{\frac{1}{n}}\).</p>
<ol type="1">
<li>
<div> </div>

\(\begin{array}{cccc}&amp; &amp; &amp; \sqrt{5y}\hfill \\ \begin{array}{c}\text{No index is shown, so it is 2.}\hfill \\ \text{The denominator of the exponent will be 2.}\hfill \end{array}\hfill &amp; &amp; &amp; {\left(5y\right)}^{\frac{1}{2}}\hfill \end{array}\)</li>
<li>
<div> </div>

\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{1em}{0ex}}\sqrt[3]{4x}\hfill \\ \begin{array}{c}\text{The index is 3, so the denominator of the}\hfill \\ \text{exponent is 3.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{1em}{0ex}}{\left(4x\right)}^{\frac{1}{3}}\hfill \end{array}\)</li>
<li>
<div> </div>

\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{1em}{0ex}}3\sqrt[4]{5z}\hfill \\ \begin{array}{c}\text{The index is 4, so the denominator of the}\hfill \\ \text{exponent is 4.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{1em}{0ex}}3{\left(5z\right)}^{\frac{1}{4}}\hfill \end{array}\)</li>
</ol>
</div>
</div>
</div>
</div>
</div>
<p>In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<p>(a) \({25}^{\frac{1}{2}}\)<br/>
(b) \({64}^{\frac{1}{3}}\)<br/>
(c) \({256}^{\frac{1}{4}}\).</p>
</div>
<h3>Solution</h3>
<div>(a)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{7em}{0ex}}{25}^{\frac{1}{2}}\hfill \\ \text{Rewrite as a square root.}\hfill &amp; &amp; &amp; \phantom{\rule{7em}{0ex}}\sqrt{25}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{7em}{0ex}}5\hfill \end{array}\)</p>
<div>(b)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{4em}{0ex}}{64}^{\frac{1}{3}}\hfill \\ \text{Rewrite as a cube root.}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\sqrt[3]{64}\hfill \\ \text{Recognize 64 is a perfect cube.}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\sqrt[3]{{4}^{3}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}4\hfill \end{array}\)</p>
<div>(c)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; {256}^{\frac{1}{4}}\hfill \\ \text{Rewrite as a fourth root.}\hfill &amp; &amp; &amp; \sqrt[4]{256}\hfill \\ \text{Recognize 256 is a perfect fourth power.}\hfill &amp; &amp; &amp; \sqrt[4]{{4}^{4}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; 4\hfill \end{array}\)</p>
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</div>
</div>
</div>
<p>Be careful of the placement of the negative signs in the next example. We will need to use the property \({a}^{\text{−}n}=\frac{1}{{a}^{n}}\) in one case.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<p>(a) \({\left(-64\right)}^{\frac{1}{3}}\)<br/>
(b) \(\text{−}{64}^{\frac{1}{3}}\)<br/>
(c) \({\left(64\right)}^{-\frac{1}{3}}\).</p>
</div>
<h3>Solution</h3>
<div>(a)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{1em}{0ex}}{\left(-64\right)}^{\frac{1}{3}}\hfill \\ \text{Rewrite as a cube root.}\hfill &amp; &amp; &amp; \phantom{\rule{1em}{0ex}}\sqrt[3]{-64}\hfill \\ \text{Rewrite}\phantom{\rule{0.2em}{0ex}}-64\phantom{\rule{0.2em}{0ex}}\text{as a perfect cube.}\hfill &amp; &amp; &amp; \phantom{\rule{1em}{0ex}}\sqrt[3]{{\left(-4\right)}^{3}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{1em}{0ex}}-4\hfill \end{array}\)</p>
<div>(b)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \text{−}{64}^{\frac{1}{3}}\hfill \\ \text{The exponent applies only to the 64.}\hfill &amp; &amp; &amp; \text{−}\left({64}^{\frac{1}{3}}\right)\hfill \\ \text{Rewrite as a cube root.}\hfill &amp; &amp; &amp; \text{−}\sqrt[3]{64}\hfill \\ \text{Rewrite 64 as}\phantom{\rule{0.2em}{0ex}}{4}^{3}.\hfill &amp; &amp; &amp; \text{−}\sqrt[3]{{4}^{3}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; -4\hfill \end{array}\)</p>
<div>(c)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{4em}{0ex}}{\left(64\right)}^{-\frac{1}{3}}\hfill \\ \begin{array}{c}\text{Rewrite as a fraction with}\hfill \\ \text{a positive exponent, using}\hfill \\ \text{the property,}\phantom{\rule{0.2em}{0ex}}{a}^{\text{−}n}=\frac{1}{{a}^{n}}.\hfill \\ \text{Write as a cube root.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{1}{\sqrt[3]{64}}\hfill \\ \text{Rewrite 64 as}\phantom{\rule{0.2em}{0ex}}{4}^{3}.\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{1}{\sqrt[3]{{4}^{3}}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{1}{4}\hfill \end{array}\)</p>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<p>(a) \({\left(-16\right)}^{\frac{1}{4}}\)<br/>
(b) \(\text{−}{16}^{\frac{1}{4}}\)<br/>
(c) \({\left(16\right)}^{-\frac{1}{4}}\).</p>
</div>
<h3>Solution</h3>
<div>(a)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; {\left(-16\right)}^{\frac{1}{4}}\hfill \\ \text{Rewrite as a fourth root.}\hfill &amp; &amp; &amp; \sqrt[4]{-16}\hfill \\ \text{There is no real number whose fourth power is}\phantom{\rule{0.2em}{0ex}}-16.\hfill &amp; &amp; &amp; \end{array}\)</p>
<div>(b)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{6em}{0ex}}\text{−}{16}^{\frac{1}{4}}\hfill \\ \begin{array}{c}\text{The exponent only applies to the 16.}\hfill \\ \text{Rewrite as a fourth root.}\hfill \end{array}\hfill &amp; &amp; &amp; \phantom{\rule{6em}{0ex}}\text{−}\sqrt[4]{16}\hfill \\ \text{Rewrite}\phantom{\rule{0.2em}{0ex}}16\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}{2}^{4}.\hfill &amp; &amp; &amp; \phantom{\rule{6em}{0ex}}\text{−}\sqrt[4]{{2}^{4}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{6em}{0ex}}-2\hfill \end{array}\)</p>
<div>(c)</div>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{5.5em}{0ex}}{\left(16\right)}^{-\frac{1}{4}}\hfill \\ \text{Rewrite using the property}\phantom{\rule{0.2em}{0ex}}{a}^{\text{−}n}=\frac{1}{{a}^{n}}.\hfill &amp; &amp; &amp; \phantom{\rule{5.5em}{0ex}}\frac{1}{{\left(16\right)}^{\frac{1}{4}}}\hfill \\ \text{Rewrite as a fourth root.}\hfill &amp; &amp; &amp; \phantom{\rule{5.5em}{0ex}}\frac{1}{\sqrt[4]{16}}\hfill \\ \text{Rewrite}\phantom{\rule{0.2em}{0ex}}16\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}{2}^{4}.\hfill &amp; &amp; &amp; \phantom{\rule{5.5em}{0ex}}\frac{1}{\sqrt[4]{{2}^{4}}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{5.5em}{0ex}}\frac{1}{2}\hfill \end{array}\)</p>
</div>
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Simplifying Expressions With <em>a<sup>1/n</sup></em>

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Simplifying Expressions With <em>a<sup>1/n</sup></em>

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