Simplify Expressions Using the Quotient Property for Exponents

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

Summary of Exponent Properties for Multiplication

If \(text{a and b}\) are real numbers, and \(\text{m and n}\) are whole numbers, then

Product Property \(a^{m} \cdot a^{n}=a^{m+n}\)
Power Property \(\left(a^{m}\right)^{n}=a^{m \cdot n}\)
Product to a Power \((a b)^{m}=a^{m} b^{m}\)

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 \(a, b\), and \(c\) are whole numbers where \(b \neq 0, c \neq 0\),

\(\text{then} \frac{a}{b}=\frac{a \cdot c}{b \cdot c} \, \text{and} \, \frac{a \cdot c}{b \cdot c}=\frac{a}{b}\)

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

Consider \(\frac{x^{5}}{x^{2}}\) and \(\frac{x^{2}}{x^{3}}\)
What do they mean? \(\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}\)   \(\frac{x \cdot x}{x \cdot x \cdot x}\)
Use the Equivalent Fractions Property. \(\frac{\not {x} \cdot \not{x} \cdot x \cdot x \cdot x}{\not \not{x} \cdot \not{x}}\)   \(\frac{\not {x} \cdot \not{x} \cdot 1 }{\not \not{x} \cdot \not{x} \cdot \not{x}}\)
Simplify. \(x^{3}\)   \(\frac{1}{x}\)

 

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:

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

This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If \(a\) is a real number, \(a \neq 0\), and \(m\) and \(n\) are whole numbers, then

\(\frac{a^{m}}{a^{n}}=a^{m-n}, m>n\) and \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}, n>m\)

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

\(\begin{aligned} \frac{3^{4}}{3^{2}} &=3^{4-2} & \frac{5^{2}}{5^{3}} &=\frac{1}{5^{3-2}} \\ \frac{81}{9} &=3^{2} & \frac{25}{125} &=\frac{1}{5^{1}} \\ 9 &=9 \text{✓} & \frac{1}{5} &=\frac{1}{5} \text{✓} \end{aligned}\)

Example 6.59

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. \(\frac{x^{9}}{x^{2}}\)
    Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=a^{m-n}\). \(X^{9-7}\)
    Simplify. \(x^{2}\)

  2. Since \(10 > 2\), there are more factors of x in the numerator. \(\frac{3^{10}}{3^{2}}\)
    Use the Quotient Property, aman=am−n. \(3^{10-2}\)
    Simplify. \(3^{8}\)


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

Try It 6.117

Simplify: 

  1. \(\frac{x^{15}}{x^{10}}\) 
  2. \(\frac{6^{14}}{6^{5}}\).
Try It 6.118

Simplify: 

  1. \(\frac{y^{43}}{y^{37}}\)
  2. \(\frac{10^{15}}{10^{7}}\).

Example 6.60

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. \(\frac{b^{s}}{b^{12}}\)
    Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(\frac{1}{b^{12-8}}\)
    Simplify. \(\frac{1}{b^{4}}\)

  2. Since \(5 > 3\), there are more factors of \(3\) in the denominator. \(\frac{7^{3}}{7^{5}}\)
    Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(\frac{1}{7^{5-3}}\)
    Simplify. \(\frac{1}{7^{2}}\)
    Simplify. \(\frac{1}{49}\)


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

Try It 6.119

Simplify: 

  1. \(\frac{x^{18}}{x^{22}}\) 
  2. \(\frac{12^{15}}{12^{30}}\).
Try It 6.120

Simplify:

  1. \(\frac{m^{7}}{m^{15}}\) 
  2. \(\frac{9^{8}}{9^{19}}\).

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 6.61

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 \(a\)'s in the denominator and so we will end up with factors in the denominator.

      \(\frac{a^{5}}{a^{9}}\)
    Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(\frac{1}{a^{9-5}}\)
    Simplify. \(\frac{1}{a^{4}}\)

  2. Notice there are more factors of \(x\) in the numerator, since \(11 > 7\). So we will end up with factors in the numerator.

      \(\frac{x^{11}}{X^{7}}\)
    Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(x^{11-1}\)
    Simplify. \(x^{4}\)
Try It 6.121

Simplify: 

  1. \(\frac{b^{19}}{b^{11}}\) 
  2. \(\frac{z^{5}}{z^{11}}\).
Try It 6.122

Simplify: 

  1. \(\frac{p^{9}}{p^{17}}\)
  2. \(\frac{w^{13}}{w^{9}}\).
Back to '7.5: Dividing Monomials'