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<h4>Rewriting Radicals</h4>
<p class="os-raise-text-bold">Example 1</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/4c149ceb45a3e32a53d2f947157509651b7c3495" width="300"></p>
<p>When writing radicals as a term with a rational exponent, the exponent becomes the numerator of the fraction in the exponent, the root or index.</p>
<p>In the example above, the 2 is the exponent and becomes the top of the fraction. The 5 is the index and becomes the bottom of the fraction.</p>
<p><strong>Step 1 -</strong> Identify the index.<br>
5</p>
<p><strong>Step 2 - </strong>Identify the exponent (inside the radical).<br>
2</p>
<p><strong>Step 3 -</strong> Write the power over the root.<br>
\(8^{ \frac25}\)</p>
<p> </p>
<p>Let’s simplify a rational exponent with a variable. </p>
<p class="os-raise-text-bold">Example 2</p>
<p>\(\sqrt[4]{5x^3}\)</p>
<p dir="ltr"><strong>Step 1 -</strong> Identify the index.<br>
4</p>
<br />
<p dir="ltr"><strong>Step 2 -</strong> Identify the exponent (inside the radical).<br>
3</p>
<strong><br />
</strong>
<p dir="ltr"><strong>Step 3 -</strong> Write the power over the root.<br>
\(5x^{\frac34}\)</p>
<p> </p>
<h4>Try It: Rewriting Radicals</h4>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<li>Write \(\sqrt[3]{12^2}\) as a term with a rational exponent.</li>
</ol>
<p> </p>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p>Here is how to rewrite the radical expression:</p>
<br>
<p><strong>Step 1 - </strong>Identify the index.<br>
3</p>
<p><strong>Step 2 -</strong> Identify the exponent (inside the radical).<br>
2</p>
<p><strong>Step 3 -</strong> Write the power over the root.<br>
\(12^{\frac23}\)</p>
<br>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>Write \(\sqrt[6]{x^5}\) as a term with a rational exponent.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2">
<p>Compare your answer:</p>
<p>Here is how to rewrite the radical expression:</p>
<br>
<p><strong>Step 1 -</strong> Identify the index.<br>
6</p>
<p><strong>Step 2 -</strong> Identify the exponent (inside the radical).<br>
5</p>
<p><strong>Step 3-</strong> Write the power over the root.<br>
\(x^\frac56\)</p>
<br>
</div>