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<h2>Simplifying Expressions with Integer Exponents</h2>
<p>All the exponent properties we developed earlier in this tutorial with whole number exponents apply to integer exponents, too. We restate them here for reference.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Definition: Summary of Exponent Properties</h3>
<p>If \(a,b\) are real numbers and \(m,n\) are integers, then</p>
<p>\(\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill &amp; &amp; &amp; {a}^{m}·{a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill &amp; &amp; &amp; {\left({a}^{m}\right)}^{n}={a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill &amp; &amp; &amp; {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill &amp; &amp; &amp; \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill &amp; &amp; &amp; {a}^{0}=1,\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill &amp; &amp; &amp; {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.2em}{0ex}}b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill &amp; &amp; &amp; {a}^{-n}=\frac{1}{{a}^{n}}\hfill \end{array}\)</p>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<ol type="1">
<li>\(\phantom{\rule{0.2em}{0ex}}{x}^{-4}·{x}^{6}\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}{y}^{-6}·{y}^{4}\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}{z}^{-5}·{z}^{-3}\)</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td style="text-align: center;">\({x}^{-4}·{x}^{6}\)</td>
</tr>
<tr>
<td>Use the Product Property, \({a}^{m}·{a}^{n}={a}^{m+n}.\)</td>
<td style="text-align: center;">\({x}^{-4+6}\)</td>
</tr>
<tr>
<td>Simplify.</td>
<td style="text-align: center;">\({x}^{2}\)</td>
</tr>
</tbody>
</table>
<table data-label="" summary=".">
<tbody>
<tr>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td style="text-align: center;">\({y}^{-6}·{y}^{4}\)</td>
</tr>
<tr>
<td>The bases are the same, so add the exponents.</td>
<td style="text-align: center;">\({y}^{-6+4}\)</td>
</tr>
<tr>
<td>Simplify.</td>
<td style="text-align: center;">\({y}^{-2}\)</td>
</tr>
<tr>
<td>Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)</td>
<td style="text-align: center;">\(\frac{1}{{y}^{2}}\)</td>
</tr>
</tbody>
</table>
<table data-label="" summary=".">
<tbody>
<tr>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td style="text-align: center;">\({z}^{-5}·{z}^{-3}\)</td>
</tr>
<tr>
<td>The bases are the same, so add the exponents.</td>
<td style="text-align: center;">\({z}^{-5-3}\)</td>
</tr>
<tr>
<td>Simplify.</td>
<td style="text-align: center;">\({z}^{-8}\)</td>
</tr>
<tr>
<td>Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)</td>
<td style="text-align: center;">\(\frac{1}{{z}^{8}}\)</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>In the next two examples, we’ll start by using the <strong>Commutative Property</strong> to group the same variables together. This makes it easier to identify the like bases before using the <strong>Product Property of Exponents</strong>.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify: \(\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right).\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td> </td>
<td style="text-align: center;">\(\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right)\)</td>
</tr>
<tr>
<td>Use the Commutative Property to get like bases together.</td>
<td style="text-align: center;">\({m}^{4}{m}^{-5}·{n}^{-2}{n}^{-3}\)</td>
</tr>
<tr>
<td>Add the exponents for each base.</td>
<td style="text-align: center;">\({m}^{-1}·{n}^{-5}\)</td>
</tr>
<tr>
<td>Take reciprocals and change the signs of the exponents.</td>
<td style="text-align: center;">\(\frac{1}{{m}^{1}}·\frac{1}{{n}^{5}}\)</td>
</tr>
<tr>
<td>Simplify.</td>
<td style="text-align: center;">\(\frac{1}{m{n}^{5}}\)</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>If the monomials have numerical coefficients, we multiply the coefficients, just as we did earlier.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify: \(\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right).\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td> </td>
<td style="text-align: center;">\(\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)\)</td>
</tr>
<tr>
<td>Rewrite with the like bases together.</td>
<td style="text-align: center;">\(2\left(-5\right)·\left({x}^{-6}{x}^{5}\right)·\left({y}^{8}{y}^{-3}\right)\)</td>
</tr>
<tr>
<td>Simplify.</td>
<td style="text-align: center;">\(-10·{x}^{-1}·{y}^{5}\)</td>
</tr>
<tr>
<td>Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)</td>
<td style="text-align: center;">\(-10·\frac{1}{{x}^{1}}·{y}^{5}\)</td>
</tr>
<tr>
<td>Simplify.</td>
<td style="text-align: center;">\(\frac{-10{y}^{5}}{x}\)</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>In the next two examples, we’ll use the <strong>Power Property</strong> and the <strong>Product to a Power Property</strong>.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify: \({\left({k}^{3}\right)}^{-2}.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td> </td>
<td style="text-align: center;">\({\left({k}^{3}\right)}^{-2}\)</td>
</tr>
<tr>
<td>Use the Product to a Power Property, \({\left(ab\right)}^{m}={a}^{m}{b}^{m}.\)</td>
<td style="text-align: center;">\({k}^{3\left(-2\right)}\)</td>
</tr>
<tr>
<td>Simplify.</td>
<td style="text-align: center;">\({k}^{-6}\)</td>
</tr>
<tr>
<td>Rewrite with a positive exponent.</td>
<td style="text-align: center;">\(\frac{1}{{k}^{6}}\)</td>
</tr>
</tbody>
</table>
</div>
</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: \({\left(5{x}^{-3}\right)}^{2}.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td> </td>
<td style="text-align: center;">\({\left(5{x}^{-3}\right)}^{2}\)</td>
</tr>
<tr>
<td>Use the Product to a Power Property, \({\left(ab\right)}^{m}={a}^{m}{b}^{m}.\)</td>
<td style="text-align: center;">\({5}^{2}{\left({x}^{-3}\right)}^{2}\)</td>
</tr>
<tr>
<td>Simplify \({5}^{2}\) and multiply the exponents of \(x\) using the

<div> </div>

Power Property, \({\left({a}^{m}\right)}^{n}={a}^{m·n}.\)</td>
<td style="text-align: center;">\(25{k}^{-6}\)</td>
</tr>
<tr>
<td>Rewrite \({x}^{-6}\) by using the definition of a negative

<div> </div>

exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)</td>
<td style="text-align: center;">\(25·\frac{1}{{x}^{6}}\)</td>
</tr>
<tr>
<td>Simplify</td>
<td style="text-align: center;">\(\frac{25}{{x}^{6}}\)</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>To simplify a fraction, we use the <strong>Quotient Property</strong>.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify: \(\frac{{r}^{5}}{{r}^{-4}}.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" style="width: 491px;" summary="r to the 5th over r to the negative 4 is shown. The first step says, “Use the Quotient Property, a to the m over a to the n equals a to the m minus n.” Beside that is r to the 5 minus red negative 4. It says, “Be careful to subtract 5 minus red negative 4.” The last line says to simplify and shows r to the 9th.">
<tbody>
<tr>
<td style="width: 374px;"> </td>
<td style="width: 105px;"><img alt="." src="https://data.ulearngo.com/assets/f103b210-a5ef-43e7-9eff-62f1fc367a69"/></td>
</tr>
<tr>
<td style="width: 374px;">Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).</td>
<td style="width: 105px;"><img alt="." src="https://data.ulearngo.com/assets/bce6fa44-6696-479a-997a-71711b352232"/></td>
</tr>
<tr>
<td style="width: 374px;"><img alt="." src="https://data.ulearngo.com/assets/f4fa294f-2b61-4fdd-9d0c-b7317d0a44dd"/></td>
<td style="width: 105px;"> </td>
</tr>
<tr>
<td style="width: 374px;">Simplify.</td>
<td style="width: 105px;"><img alt="." src="https://data.ulearngo.com/assets/4fae3dbf-4561-481c-af8b-2d4811854b8b"/></td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>

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Simplifying Expressions with Integer Exponents

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

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Discover how to identify, determine the degree of, add, subtract, multiply, divide, and factor polynomials while also simplifying expressions with exponents in this tutorial.


Simplifying Expressions with Integer Exponents

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