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<h4>Activity (20 minutes)</h4>
<p>This activity prompts students to explore the parts of a radical as well as radicals with indexes above 2. Students
may need a review of what an exponent does to a value, especially with negative numbers involved.</p>
<h4>Launch</h4>
<p>Arrange students into groups of 2.</p>
<br>
<h4>Student Activity</h4>
<p>Work with a partner to discuss and answer the following questions.</p>
<ol class="os-raise-noindent">
<li> Complete the table when \(n=-2\) on your paper and check your answer. </li>
</ol>
<br>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">Expression</th>
<th scope="col">Process</th>
<th scope="col">Simplified Result</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(n\)</td>
<td>\((-2)\)</td>
<td>-2</td>
</tr>
<tr>
<td>\(n^2\)</td>
<td>\((-2)^2=(-2)(-2)\)</td>
<td>4</td>
</tr>
<tr>
<td>\(n^3\)</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>\(n^4\)</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>\(n^5\)</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
<p class="os-raise-text-bold">Answer:</p>
<br>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">Expression</th>
<th scope="col">Process</th>
<th scope="col">Simplified Result</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(n\)</td>
<td>\((-2)\)</td>
<td>-2</td>
</tr>
<tr>
<td>\(n^2\)</td>
<td>\((-2)^2=(-2)(-2)\)</td>
<td>4</td>
</tr>
<tr>
<td>\(n^3\)</td>
<td>\((-2)^3=(-2)(-2)(-2)\)</td>
<td>-8</td>
</tr>
<tr>
<td>\(n^4\)</td>
<td>\((-2)^4\)</td>
<td>16</td>
</tr>
<tr>
<td>\(n^5\)</td>
<td>\((-2)^5\)</td>
<td>-32</td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent" start="2">
<li> What do you notice about the answers when the powers are even? <br>
<br>
<strong>Answer:</strong> They are positive. </li>
</ol>
<ol class="os-raise-noindent" start="3">
<li> What do you notice about the answers when the powers are odd? <br>
<br>
<strong>Answer:</strong> They are negative. </li>
</ol>
<p>The \(\sqrt a\), or square root of \(a\), is the same as \(\sqrt[2]a\). What about when we want to take roots that
are higher than the square root? These are called <span>radicals</span>.</p>
<p>The nth root, or radical, is written \(\sqrt[n]a\). If \(b^n=a\), then \(b\) is an nth root of \(a\). The \(n\) is
called the <span>index</span> of the radical. Just like \(\sqrt[2]n\) is called the square root, \(\sqrt[3]n\) is the
cubed root, \(\sqrt[4]n\) is the fourth root, and so on. The number inside the radical is called the <span>radicand</span>.</p>
<ol class="os-raise-noindent" start="4">
<li> Using what you did with the powers in number 1, complete the table to find the roots of each. </li>
</ol>
<br>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">\(x^y = z\)</th>
<th scope="col">\(\sqrt[y]{z}=x\)</th>
</tr>
</thead>
<tbody>
<tr>
<td> \(4^3\;=\;64\) </td>
<td>\(\sqrt[3]{64}\;=\;4\)</td>
</tr>
<tr>
<td> \(3^4\;=\;81\) </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> \(\sqrt[5]{-32}\;=\;-2\) </td>
</tr>
</tbody>
</table>
<br>
<p class="os-raise-text-bold">Answer:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">\(x^y = z\)</th>
<th scope="col">\(\sqrt[y]{z}=x\)</th>
</tr>
</thead>
<tbody>
<tr>
<td> \(4^3\;=\;64\) </td>
<td>\(\sqrt[3]{64}\;=\;4\)</td>
</tr>
<tr>
<td> \(3^4\;=\;81\) </td>
<td>\(\sqrt[4]{81}\;=\;3\)</td>
</tr>
<tr>
<td>\({(-2)}^5\;=\;-32\)</td>
<td> \(\sqrt[5]{-32}\;=\;-2\) </td>
</tr>
</tbody>
</table>
<br>
<p>Could we have an even root of a negative number? We know that the square root of a negative number is not a real
number. The same is true for any even root. <em>Even</em> roots of negative numbers are not real numbers. <em>Odd</em> roots of negative numbers are real numbers.</p>
<h4>Anticipated Misconceptions</h4>
<p>Students may have trouble with multiplication of the negative signs and the rules associated with it. If needed, add
another column to the table for students in their notebook to allow them to expand each term to find the answer, and
then take the root.</p>
<h4>Activity Synthesis</h4>
<p>Invite students to share how they arrived at their answers.</p>
<h3>5.2.2: Self Check</h3>
<p class="os-raise-text-bold os-raise-text-italicize">After the activity, students will answer the following question to check their
understanding of the concepts explored in the activity.</p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Simplify \(\sqrt[3]{-27}\).</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td> 3 </td>
<td> Incorrect. Let’s try again a different way: Think about what number is used as a factor 3 times to get
–27. Watch the signs of the factor. The answer is –3. </td>
</tr>
<tr>
<td> 9 </td>
<td> Incorrect. Let’s try again a different way: Think about what number is used as a factor 3 times to get
–27. The answer is –3. </td>
</tr>
<tr>
<td>–3 </td>
<td> That’s correct! Check yourself: \((-3)\;\times\;(-3)\;\times\;(-3)=-27\), so \(\sqrt[3]{-27}\;=\;-3\). </td>
</tr>
<tr>
<td>–9 </td>
<td> Incorrect. Let’s try again a different way: This would be the result of dividing –27 by 3. Think
about what number is used as a factor 3 times to get –27. The answer is –3. </td>
</tr>
</tbody>
</table>
<br>
<h3>5.2.2: Additional Resources </h3>
<p class="os-raise-text-bold os-raise-text-italicize">The following content is available to students who would like more support based on their experience with
the self check. Students will not automatically have access to this content, so you may wish to share it with
those who could benefit from it.</p>
<br>
<h4>Simplifying Radicals</h4>
<p><img src="https://k12.openstax.org/contents/raise/resources/71116f4837c799d5fae93c3319f60cabde089e6f" width="300"> </p>
<p>When simplifying <span>radicals</span>, think about which number will be used as a factor the number of times
indicated by the <span>index</span> to get the number under the radical.</p>
<p>In the example above,</p>
<p>\(2\;\times\;2\;\times\;2=\;8\), so \(\sqrt[3]8\;=\;2\).</p>
<p>Remember, when the index is odd, the root is allowed to be negative. When the index is even, the root cannot be
negative.</p>
<h4>Try It: Simplifying Radicals</h4>
<p>Simplify.</p>
<p>\(\sqrt[5]{-32}\;\)</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to simplify a radical:</p>
<br>
<p><strong>Step 1 -</strong> Identify the index.<br>
5</p>
<p><strong>Step 2 -</strong> Identify the radicand (inside the radical).<br>
–32</p>
<p><strong>Step 3 -</strong> What number can be used as a factor 5 times to get –32?<br>
\((-2)\;\times\;(-2)\;\times\;(-2)\;\times\;(-2)\;\times\;(-2)=-32\)</p>
<p><strong>Step 4 -</strong> Simplify.<br>
\(\sqrt[5]{-32}\;=\;-2\)</p>