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Simplifying Expressions Using the Quotient Property For Exponents

Simplifying Expressions Using the Quotient Property for Exponents

Earlier in this tutorial, we developed the properties of exponents for multiplication. We summarize these properties below.

Summary of Exponent Properties for Multiplication

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

\(\begin{array}{cccccc}\mathbf{\text{Product Property}}\hfill & & & \hfill {a}^{m}·{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \end{array}\)

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.

Equivalent Fractions Property

If \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\) are whole numbers where \(b\ne 0,c\ne 0\),

\(\text{then}\phantom{\rule{1em}{0ex}}\frac{a}{b}=\frac{a·c}{b·c}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{a·c}{b·c}=\frac{a}{b}\)

As before, we’ll try to discover a property by looking at some examples.

\(\begin{array}{cccccccccc}\text{Consider}\hfill & & & \hfill \frac{{x}^{5}}{{x}^{2}}\hfill & & & \hfill \text{and}\hfill & & & \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ \text{What do they mean?}\hfill & & & \hfill \frac{x·x·x·x·x}{x·x}\hfill & & & & & & \hfill \frac{x·x}{x·x·x}\hfill \\ \text{Use the Equivalent Fractions Property.}\hfill & & & \hfill \frac{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}·x·x·x}{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}}\hfill & & & & & & \hfill \frac{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}·1}{\require{cancel}\cancel{x}·\require{cancel}\cancel{x}·x}\hfill \\ \text{Simplify.}\hfill & & & \hfill {x}^{3}\hfill & & & & & & \hfill \frac{1}{x}\hfill \end{array}\)

Notice, in each case the bases were the same and we subtracted exponents.

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.

We write:

\(\begin{array}{cccc}\hfill \frac{{x}^{5}}{{x}^{2}}\hfill & & & \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ \hfill {x}^{5-2}\hfill & & & \hfill \frac{1}{{x}^{3-2}}\hfill \\ \hfill {x}^{3}\hfill & & & \hfill \frac{1}{x}\hfill \end{array}\)

This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

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

\(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},n>m\)

A couple of examples with numbers may help to verify this property.

\(\begin{array}{cccccccc}\hfill \frac{{3}^{4}}{{3}^{2}}& =\hfill & {3}^{4-2}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{5}^{2}}{{5}^{3}}& =\hfill & \frac{1}{{5}^{3-2}}\hfill \\ \hfill \frac{81}{9}& =\hfill & {3}^{2}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{25}{125}& =\hfill & \frac{1}{{5}^{1}}\hfill \\ \hfill 9& =\hfill & 9✓\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{5}& =\hfill & \frac{1}{5}✓\hfill \end{array}\)

Example

Simplify:

  1. \(\frac{{x}^{9}}{{x}^{7}}\)
  2. \(\frac{{3}^{10}}{{3}^{2}}.\)

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

  1.  
    Since 9 > 7, there are more factors of x in the numerator.x to the ninth power divided by x to the seventh power.
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).x to the power of 9 minus 7.
    Simplify.x squared.
  2.  
    Since 10 > 2, there are more factors of x in the numerator.3 to the tenth power divided by 3 squared.
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).3 to the power of 10 minus 2.
    Simplify.3 to the eighth power.
     

    Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

Example

Simplify:

  1. \(\frac{{b}^{8}}{{b}^{12}}\)
  2. \(\frac{{7}^{3}}{{7}^{5}}.\)

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

  1.  
    Since 12 > 8, there are more factors of b in the denominator.b to the eighth power divided b to the twelfth power.
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).1 divided by b to the power of 12 minus 8.
    Simplify.1 divided by b to the fourth power.
  2.  
    Since 5 > 3, there are more factors of 3 in the denominator.7 cubed divided by 7 to the fifth power.
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).1 divided by 7 to the power of 5 minus 3.
    Simplify.1 divided by 7 squared.
    Simplify.1 forty-ninth.
     

    Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

Notice the difference in the two previous examples:

  • If we start with more factors in the numerator, we will end up with factors in the numerator.
  • If we start with more factors in the denominator, we will end up with factors in the denominator.

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.

Example

Simplify:

  1. \(\frac{{a}^{5}}{{a}^{9}}\)
  2. \(\frac{{x}^{11}}{{x}^{7}}.\)

Solution

  1. Is the exponent of \(a\) larger in the numerator or denominator? Since 9 > 5, there are more \(a\text{‘}\text{s}\) in the denominator and so we will end up with factors in the denominator.
     a to the fifth power divided by a to the ninth power.
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).1 divided by a to the power of 9 minus 5.
    Simplify.1 divided by a to the fourth power.
  2. Notice there are more factors of \(x\) in the numerator, since 11 > 7. So we will end up with factors in the numerator.
     x to the eleventh power divided by x to the seventh power.
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).x to the power of 11 minus 7.
    Simplify.x to the fourth power.

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