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

Before you get started, take this readiness quiz.

  1. Simplify: 3 4 · 3 4 .
    If you missed this problem, review [link] .
  2. Simplify: ( −2 ) ( −2 ) ( −2 ) .
    If you missed this problem, review [link] .

Simplify expressions with exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 2 4 means to multiply 2 by itself 4 times, so 2 4 means 2 · 2 · 2 · 2 .

Let’s review the vocabulary for expressions with exponents.

Exponential notation

This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.

This is read a to the m t h power.

In the expression a m , the exponent m tells us how many times we use the base a as a factor.

This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Simplify: 4 3 7 1 ( 5 6 ) 2 ( 0.63 ) 2 .

Solution


4 3 Multiply three factors of 4. 4 · 4 · 4 Simplify. 64


7 1 Multiply one factor of 7. 7


( 5 6 ) 2 Multiply two factors. ( 5 6 ) ( 5 6 ) Simplify. 25 36


( 0.63 ) 2 Multiply two factors. ( 0.63 ) ( 0.63 ) Simplify. 0.3969

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Simplify: 6 3 15 1 ( 3 7 ) 2 ( 0.43 ) 2 .

216 15 9 49 0.1849

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Simplify: 2 5 21 1 ( 2 5 ) 3 ( 0.218 ) 2 .

32 21 8 125 0.047524

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Simplify: ( −5 ) 4 5 4 .

Solution


  1. ( −5 ) 4 Multiply four factors of −5 . ( −5 ) ( −5 ) ( −5 ) ( −5 ) Simplify. 625


  2. 5 4 Multiply four factors of 5. ( 5 · 5 · 5 · 5 ) Simplify. −625
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Simplify: ( −3 ) 4 3 4 .

81 −81

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Simplify: ( −13 ) 2 13 2 .

169 −169

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Notice the similarities and differences in [link] and [link] ! Why are the answers different? As we follow the order of operations in part the parentheses tell us to raise the ( −5 ) to the 4 th power. In part we raise just the 5 to the 4 th power and then take the opposite.

Simplify expressions using the product property for exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

We’ll derive the properties of exponents by looking for patterns in several examples.

First, we will look at an example that leads to the Product Property.

x squared times x cubed.
What does this mean?
How many factors altogether?
x times x, multiplied by x times x. x times x has two factors. x times x times x has three factors. 2 plus 3 is five factors.
So, we have x to the fifth power.
Notice that 5 is the sum of the exponents, 2 and 3. x squared times x cubed is x to the power of 2 plus 3, or x to the fifth power.

We write:

x 2 · x 3 x 2 + 3 x 5

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents .

Product property for exponents

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

a m · a n = a m + n

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

2 2 · 2 3 = ? 2 2 + 3 4 · 8 = ? 2 5 32 = 32

Simplify: y 5 · y 6 .

Solution

y to the fifth power times y to the sixth power.
Use the product property, a m · a n = a m+n . y to the power of 5 plus 6.
Simplify. y to the eleventh power.

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Simplify: b 9 · b 8 .

b 17

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Simplify: x 12 · x 4 .

x 16

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Simplify: 2 5 · 2 9 3 · 3 4 .

Solution


  1. 2 to the fifth power times 2 to the ninth power.
    Use the product property, a m · a n = a m+n . 2 to the power of 5 plus 9.
    Simplify. 2 to the 14th power.

  2. 3 to the fifth power times 3 to the fourth power.
    Use the product property, a m · a n = a m+n . 3 to the power of 5 plus 4.
    Simplify. 3 to the ninth power.
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Simplify: 5 · 5 5 4 9 · 4 9 .

5 6 4 18

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Simplify: 7 6 · 7 8 10 · 10 10 .

7 14 10 11

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Simplify: a 7 · a x 27 · x 13 .

Solution


  1. a to the seventh power times a.
    Rewrite, a = a 1 . a to the seventh power times a to the first power.
    Use the product property, a m · a n = a m+n . a to the power of 7 plus 1.
    Simplify. a to the eighth power.

  2. x to the twenty-seventh power times x to the thirteenth power.
    Notice, the bases are the same, so add the exponents. x to the power of 27 plus 13.
    Simplify. x to the fortieth power.
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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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