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By the end of this section, you will be able to:
  • Simplify expressions using the Quotient Property for Exponents
  • Simplify expressions with zero exponents
  • Simplify expressions using the quotient to a Power Property
  • Simplify expressions by applying several properties
  • Divide monomials

Before you get started, take this readiness quiz.

  1. Simplify: 8 24 .
    If you missed this problem, review [link] .
  2. Simplify: ( 2 m 3 ) 5 .
    If you missed this problem, review [link] .
  3. Simplify: 12 x 12 y .
    If you missed this problem, review [link] .

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 a and b are real numbers, and m and n are whole numbers, then

Product Property a m · a n = a m + n Power Property ( a m ) n = a m · 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 0 , c 0 ,

then a b = a · c b · c and a · c b · c = a b

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

Consider x 5 x 2 and x 2 x 3 What do they mean? x · x · x · x · x x · x x · x x · x · x Use the Equivalent Fractions Property. x · x · x · x · x x · x x · x · 1 x · x · x Simplify. x 3 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:

x 5 x 2 x 2 x 3 x 5 2 1 x 3 2 x 3 1 x

This leads to the Quotient Property for Exponents .

Quotient property for exponents

If a is a real number, a 0 , and m and n are whole numbers, then

a m a n = a m n , m > n and a m a n = 1 a n m , n > m

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

3 4 3 2 = 3 4 2 5 2 5 3 = 1 5 3 2 81 9 = 3 2 25 125 = 1 5 1 9 = 9 1 5 = 1 5

Simplify: x 9 x 7 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, 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, 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.
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Simplify: x 15 x 10 6 14 6 5 .

x 5 6 9

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Simplify: y 43 y 37 10 15 10 7 .

y 6 10 8

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Simplify: b 8 b 12 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, a m a n = 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, a m a n = 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.
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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