<h2>Simplifying Expressions Using the Quotient Property for Exponents</h2>
<p>Earlier in this tutorial, we developed the properties of exponents for multiplication. We summarize these properties below.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Summary of Exponent Properties for Multiplication</h3>
<p>If \(a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\) are real numbers, and \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then</p>
<p>\(\begin{array}{cccccc}\mathbf{\text{Product Property}}\hfill &amp; &amp; &amp; \hfill {a}^{m}·{a}^{n}&amp; =\hfill &amp; {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill &amp; &amp; &amp; \hfill {\left({a}^{m}\right)}^{n}&amp; =\hfill &amp; {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill &amp; &amp; &amp; \hfill {\left(ab\right)}^{m}&amp; =\hfill &amp; {a}^{m}{b}^{m}\hfill \end{array}\)</p>
</div>
</div>
<p>Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Equivalent Fractions Property</h3>
<p>If \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\) are whole numbers where \(b\ne 0,c\ne 0\),</p>
<p>\(\text{then}\phantom{\rule{1em}{0ex}}\frac{a}{b}=\frac{a·c}{b·c}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{a·c}{b·c}=\frac{a}{b}\)</p>
</div>
</div>
<p>As before, we’ll try to discover a property by looking at some examples.</p>
<p>\(\begin{array}{cccccccccc}\text{Consider}\hfill &amp; &amp; &amp; \hfill \frac{{x}^{5}}{{x}^{2}}\hfill &amp; &amp; &amp; \hfill \text{and}\hfill &amp; &amp; &amp; \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ \text{What do they mean?}\hfill &amp; &amp; &amp; \hfill \frac{x·x·x·x·x}{x·x}\hfill &amp; &amp; &amp; &amp; &amp; &amp; \hfill \frac{x·x}{x·x·x}\hfill \\ \text{Use the Equivalent Fractions Property.}\hfill &amp; &amp; &amp; \hfill \frac{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}·x·x·x}{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}}\hfill &amp; &amp; &amp; &amp; &amp; &amp; \hfill \frac{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}·1}{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}·x}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \hfill {x}^{3}\hfill &amp; &amp; &amp; &amp; &amp; &amp; \hfill \frac{1}{x}\hfill \end{array}\)</p>
<p>Notice, in each case the bases were the same and we subtracted exponents.</p>
<p>When the larger exponent was in the numerator, we were left with factors in the numerator.</p>
<p>When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.</p>
<p>We write:</p>
<div>\(\begin{array}{cccc}\hfill \frac{{x}^{5}}{{x}^{2}}\hfill &amp; &amp; &amp; \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ \hfill {x}^{5-2}\hfill &amp; &amp; &amp; \hfill \frac{1}{{x}^{3-2}}\hfill \\ \hfill {x}^{3}\hfill &amp; &amp; &amp; \hfill \frac{1}{x}\hfill \end{array}\)</div>
<p>This leads to the <em>Quotient Property for Exponents</em>.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Quotient Property for Exponents</h3>
<p>If \(a\) is a real number, \(a\ne 0\), and \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then</p>
<p>\(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m&gt;n\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},n&gt;m\)</p>
</div>
</div>
<p>A couple of examples with numbers may help to verify this property.</p>
<p>\(\begin{array}{cccccccc}\hfill \frac{{3}^{4}}{{3}^{2}}&amp; =\hfill &amp; {3}^{4-2}\hfill &amp; &amp; &amp; \hfill \phantom{\rule{4em}{0ex}}\frac{{5}^{2}}{{5}^{3}}&amp; =\hfill &amp; \frac{1}{{5}^{3-2}}\hfill \\ \hfill \frac{81}{9}&amp; =\hfill &amp; {3}^{2}\hfill &amp; &amp; &amp; \hfill \phantom{\rule{4em}{0ex}}\frac{25}{125}&amp; =\hfill &amp; \frac{1}{{5}^{1}}\hfill \\ \hfill 9&amp; =\hfill &amp; 9✓\hfill &amp; &amp; &amp; \hfill \phantom{\rule{4em}{0ex}}\frac{1}{5}&amp; =\hfill &amp; \frac{1}{5}✓\hfill \end{array}\)</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<ol>
<li>\(\frac{{x}^{9}}{{x}^{7}}\)</li>
<li>\(\frac{{3}^{10}}{{3}^{2}}.\)</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<p>To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.</p>
<ol type="1">
<li>
<div> </div>
<table summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the instructions in the left column say “Since 9 is greater than 7, there are more factors of x in the numerator.” In the right column is x to the ninth power divided by x to the seventh power. One line down, the instructions in the left column say “Use Quotient Property,” which is defined as a to the m power over a to the n power equals a to the m minus n power. The right column contains x to the nine minus seventh power. One line down, the left column says “Simplify.” The right column contains x squared.">
<tbody>
<tr>
<td>Since 9 &gt; 7, there are more factors of <em>x</em> in the numerator.</td>
<td><img alt="x to the ninth power divided by x to the seventh power." src="https://data.ulearngo.com/assets/0d41f0b3-0270-44d0-97d8-ca16a310d1b9"/></td>
</tr>
<tr>
<td>Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).</td>
<td><img alt="x to the power of 9 minus 7." src="https://data.ulearngo.com/assets/d97a5929-abc9-4925-8dd1-3ab1074f9e22"/></td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="x squared." src="https://data.ulearngo.com/assets/4e7cb33a-3a84-401a-88d4-319f446ec1a4"/></td>
</tr>
</tbody>
</table>
</li>
<li>
<div> </div>
<table summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the instructions in the left column say “Since 10 is greater than 2, there are more factors of 3 in the numerator.” In the right column is 3 to the tenth power divided by 3 squared. One line down, the instructions in the left column say “Use Quotient Property,” which is defined as a to the m power over a to the n power equals a to the m minus n power. The right column contains 3 to the ten minus second power. One line down, the left column says “Simplify.” The right column contains 3 to the eighth power.">
<tbody>
<tr>
<td>Since 10 &gt; 2, there are more factors of <em>x</em> in the numerator.</td>
<td><img alt="3 to the tenth power divided by 3 squared." src="https://data.ulearngo.com/assets/6d7d76d2-3239-4e1f-a67b-20685f20c921"/></td>
</tr>
<tr>
<td>Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).</td>
<td><img alt="3 to the power of 10 minus 2." src="https://data.ulearngo.com/assets/15a4ffd5-c240-411b-b6eb-b65504b615ab"/></td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="3 to the eighth power." src="https://data.ulearngo.com/assets/1f5685e4-f59d-45c3-b665-4645239be73f"/></td>
</tr>
</tbody>
</table>
<div> </div>

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.</li>
</ol>
</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:</p>
<ol>
<li>\(\frac{{b}^{8}}{{b}^{12}}\)</li>
<li>\(\frac{{7}^{3}}{{7}^{5}}.\)</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<p>To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.</p>
<ol type="1">
<li>
<div> </div>
<table summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the instructions in the left column say “Since 12 is greater than 8, there are more factors of b in the denominator.” In the right column is b to the eighth power divided b to the twelfth power. One line down, the instructions in the left column say “Use Quotient Property,” which is defined as a to the m power over a to the n power equals 1 divided by a to the n minus m power. The right column contains 1 divided by b to the twelve minus eighth power. One line down, the left column says “Simplify.” The right column contains 1 divided by b to the fourth power.">
<tbody>
<tr>
<td>Since 12 &gt; 8, there are more factors of <em>b</em> in the denominator.</td>
<td><img alt="b to the eighth power divided b to the twelfth power." src="https://data.ulearngo.com/assets/d370df22-a578-4ac2-960f-0ae4e6b098cc"/></td>
</tr>
<tr>
<td>Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).</td>
<td><img alt="1 divided by b to the power of 12 minus 8." src="https://data.ulearngo.com/assets/38f9bf94-36e7-42af-b955-f143c5cb42ee"/></td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="1 divided by b to the fourth power." src="https://data.ulearngo.com/assets/44f649bd-88d9-494f-aada-4cd893e2f8e9"/></td>
</tr>
</tbody>
</table>
</li>
<li>
<div> </div>
<table summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the instructions in the left column say “Since 5 is greater than 3, there are more factors of b in the denominator.” In the right column is 7 cubed divided 7 to the fifth power. One line down, the instructions in the left column say “Use Quotient Property,” which is defined as a to the m power over a to the n power equals 1 divided by a to the n minus m power. The right column contains 1 divided by 7 to the five minus third power. One line down, the left column says “Simplify.” The right column contains 1 divided by 7 squared. One line down, the left column says “Simplify” again. The right column contains 1 forty ninth.">
<tbody>
<tr>
<td>Since 5 &gt; 3, there are more factors of 3 in the denominator.</td>
<td><img alt="7 cubed divided by 7 to the fifth power." src="https://data.ulearngo.com/assets/3aa495d4-4755-484f-9383-d08cb336157f"/></td>
</tr>
<tr>
<td>Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).</td>
<td><img alt="1 divided by 7 to the power of 5 minus 3." src="https://data.ulearngo.com/assets/81cda553-cf50-44b8-b5fd-36f5166c6ba7"/></td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="1 divided by 7 squared." src="https://data.ulearngo.com/assets/eb6ef232-31bf-4880-9aa5-6f205b92f31c"/></td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="1 forty-ninth." src="https://data.ulearngo.com/assets/b0e1a9a0-46c3-40b1-b4e9-9f2ee6350000"/></td>
</tr>
</tbody>
</table>
<div> </div>

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.</li>
</ol>
</div>
</div>
</div>
</div>
</div>
<p>Notice the difference in the two previous examples:</p>
<ul data-bullet-style="bullet">
<li>If we start with more factors in the numerator, we will end up with factors in the numerator.</li>
<li>If we start with more factors in the denominator, we will end up with factors in the denominator.</li>
</ul>
<p>The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Simplify:</p>
<ol>
<li>\(\frac{{a}^{5}}{{a}^{9}}\)</li>
<li>\(\frac{{x}^{11}}{{x}^{7}}.\)</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<ol type="1">
<li>Is the exponent of \(a\) larger in the numerator or denominator? Since 9 &gt; 5, there are more \(a\text{'}\text{s}\) in the denominator and so we will end up with factors in the denominator.

<table summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains a to the fifth power divided by a to the ninth power. One line down, the instructions in the left column say “Use Quotient Property,” which is defined as a to the m power over a to the n power equals 1 divided by a to the n minus m power. The right column contains 1 divided by a to the nine minus fifth power. One line down, the left column says “Simplify.” The right column contains 1 divided by a to the fourth power.">
<tbody>
<tr>
<td> </td>
<td><img alt="a to the fifth power divided by a to the ninth power." src="https://data.ulearngo.com/assets/1550dc0c-6be3-4b74-93e4-5a0da80785b7"/></td>
</tr>
<tr>
<td>Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).</td>
<td><img alt="1 divided by a to the power of 9 minus 5." src="https://data.ulearngo.com/assets/b4a20227-a356-44f4-90c3-fc99a3788948"/></td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="1 divided by a to the fourth power." src="https://data.ulearngo.com/assets/6138448e-edab-451e-b398-baf2554450cc"/></td>
</tr>
</tbody>
</table>
</li>
<li>Notice there are more factors of \(x\) in the numerator, since 11 &gt; 7. So we will end up with factors in the numerator.

<table summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains x to the eleventh power divided by x to the seventh power. One line down, the instructions in the left column say “Use Quotient Property,” which is defined as a to the m power over a to the n power equals a to the m minus n power. The right column contains x to the eleven minus seventh power. One line down, the left column says “Simplify.” The right column contains x to the fourth power.">
<tbody>
<tr>
<td> </td>
<td><img alt="x to the eleventh power divided by x to the seventh power." src="https://data.ulearngo.com/assets/35aec07f-a5bf-4ffa-8791-0c4a7f1bc12e"/></td>
</tr>
<tr>
<td>Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).</td>
<td><img alt="x to the power of 11 minus 7." src="https://data.ulearngo.com/assets/6078f2e0-f9e4-4cbf-8dcc-44264ea1e386"/></td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="x to the fourth power." src="https://data.ulearngo.com/assets/36665378-a58f-4c4b-a2fb-7a48ea5edf0a"/></td>
</tr>
</tbody>
</table>
</li>
</ol>
</div>
</div>
</div>
</div>
</div>

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Simplifying Expressions Using the Quotient Property For 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, evaluate, add, subtract, multiply, and divide various types of polynomials, and simplify expressions using exponents and scientific notations in this tutorial.


Simplifying Expressions Using the Quotient Property For Exponents

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