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6.2: Use Multiplication Properties of Exponents

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Learning Objectives

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
Note

Before you get started, take this readiness quiz.

  1. Simplify: 3434
    If you missed this problem, review Example 1.6.13.
  2. Simplify: (2)(2)(2).
    If you missed this problem, review Example 1.5.13.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 24 means the product of 4 factors of 2, so 24 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 mth power.

In the expression am, 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.

Example 6.2.1

Simplify:

  1. 43
  2. 71
  3. (56)2
  4. (0.63)2

Solution

a.43 Multiply three factors of 4.444 Simplify. 64

b.71Multiply one factor of 7.7

c.(56)2 Multiply two factors. (56)(56) Simplify. 2536

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

Try It 6.2.2

Simplify:

  1. 63
  2. 151
  3. (37)2
  4. (0.43)2
Answer
  1. 216
  2. 15
  3. 949
  4. 0.1849
Try It 6.2.3

Simplify:

  1. 25
  2. 211
  3. (25)3
  4. (0.218)2
Answer
  1. 32
  2. 21
  3. 8125
  4. 0.047524
Example 6.2.4

Simplify:

  1. (5)4
  2. 54

Solution

  1. (5)4 Multiply four factors of 5(5)(5)(5) Simplify. 625
  2. 54 Multiply four factors of 5.(5555) Simplify. 625

Notice the similarities and differences in Example 6.2.4 part 1 and Example 6.2.4 part 2! Why are the answers different? As we follow the order of operations in part 1 the parentheses tell us to raise the (5) to the 4th power. In part 2 we raise just the 5 to the 4th power and then take the opposite.

Try It 6.2.5

Simplify:

  1. (3)4
  2. 34
Answer
  1. 81
  2. −81
Try It 6.2.6

Simplify:

  1. (13)4
  2. 134
Answer
  1. 169
  2. −169

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 x5
Notice that 5 is the sum of the exponents, 2 and 3. x2x3 is x2+3 or x5

We write: x2x3x2+3x5

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

aman=am+n

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

2322?=22+348?=2532=32

Example 6.2.7

Simplify: y5y6

Solution

  y5y6
Use the product property, aman=am+n. y5+6
Simplify. y11
Try It 6.2.8

Simplify: b9b8

Answer

b17

Try It 6.2.9

Simplify: x12x4

Answer

x16

Example 6.2.10

Simplify:

  1. 2529
  2. 334

Solution

a.

  2529
Use the product property, aman=am+n. 25+9
Simplify. 214

b.

  334
Use the product property, aman=am+n. 31+4
Simplify. 35
Try It 6.2.11

Simplify:

  1. 555
  2. 4949
Answer
  1. 56
  2. 418
Try It 6.2.12

Simplify:

  1. 7678
  2. 101010
Answer
  1. 714
  2. 1011
Example 6.2.13

Simplify:

  1. a7a
  2. x27x13

Solution

a.

  a7a
Rewrite, a=a1 a7a1
Use the product property, aman=am+n. a7+1
Simplify. a8

b.

  x27x13
Notice, the bases are the same, so add the exponents. x27+13
Simplify. x40
Try It 6.2.14

Simplify:

  1. p5p
  2. y14y29
Answer
  1. p6
  2. y43
Try It 6.2.15

Simplify:

  1. zz7
  2. b15b34
Answer
  1. z8
  2. b49

We can extend the Product Property for Exponents to more than two factors.

Example 6.2.16

Simplify: d4d5d2

Solution

  d4d5d2
Add the exponents, since bases are the same. d4+5+2
Simplify. d11
Try It 6.2.17

Simplify: x6x4x8

Answer

x18

Try It 6.2.18

Simplify: b5b9b5

Answer

b19

Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

  (x2)3
What does this mean?
How many factors altogether?
x squared cubed is x squared times x squared times x squared, which is x times x, multiplied by x times x, multiplied by x times x. x times x has two factors. Two plus two plus two is six factors.
So we have x6
Notice that 6 is the product of the exponents, 2 and 3. (x2)3 is x23 or x6

We write:

(x2)3x23x6

We multiplied the exponents. This leads to the Power Property for Exponents.

POWER PROPERTY FOR EXPONENTS

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

(am)n=amn

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

(32)3?=323(9)3?=36729=729

Example 6.2.19

Simplify:

  1. (y5)9
  2. (44)7

Solution

a.

  (y5)9
Use the power property, (am)n=amn. y59
Simplify. y45

b.

  (44)7
Use the power property. y47
Simplify. 428
Try It 6.2.20

Simplify:

  1. (b7)5
  2. (54)3
Answer
  1. b35
  2. 512
Try It 6.2.21

Simplify:

  1. (z6)9
  2. (37)7
Answer
  1. z54
  2. 349

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

 What does this mean? (2x)3 We group the like factors together. 2x2x2x How many factors of 2 and of x?222x3 Notice that each factor was raised to the power and (2x)3 is 23x3

We write:(2x)323x3

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

PRODUCT TO A POWER PROPERTY FOR EXPONENTS

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

(ab)m=ambm

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

(23)2?=223262?=4936=36

Example 6.2.22

Simplify:

  1. (9d)2
  2. (3mn)3.

Solution

a.

  (9d)2
Use Power of a Product Property, (ab)m=ambm. (9)2d2
Simplify. 81d2
b.
  (3mn)3.
Use Power of a Product Property, (ab)m=ambm. (3)3m3n3
Simplify. 27m3n3
Try It 6.2.23

Simplify:

  1. (12y)2
  2. (2wx)5
Answer
  1. 144y2
  2. 32w5x5
Try It 6.2.24

Simplify:

  1. (5wx)3
  2. (3y)3
Answer
  1. 125w3x3
  2. 27y3

Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

PROPERTIES OF EXPONENTS

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

Product Property aman=am+nPower Property (am)n=amnProduct to a Power (ab)m=ambm

All exponent properties hold true for any real numbers m and n. Right now, we only use whole number exponents.

Example 6.2.25

Simplify:

  1. (y3)6(y5)4
  2. (6x4y5)2

Solution

a.(y3)6(y5)4 Use the Power Property. y18y20 Add the exponents. y38
b.(6x4y5)2 Use the Product to a Power Property. (6)2(x4)2(y5)2 Use the Power Property. (6)2(x8)(y10)2 Simplify. 36x8y10

Try It 6.2.26

Simplify:

  1. (a4)5(a7)4
  2. (2c4d2)3
Answer
  1. a48
  2. 8c12d6
Try It 6.2.27

Simplify:

  1. (3x6y7)4
  2. (q4)5(q3)3
Answer
  1. 81x24y28
  2. q29
Example 6.2.28

Simplify:

  1. (5m)2(3m3)
  2. (3x2y)4(2xy2)3

Solution

  1. (5m)2(3m3) Raise 5m to the second power. 52m23m3 Simplify. 25m23m3 Use the Commutative Property. 253m2m3 Multiply the constants and add the exponents. 75m5
  2. (3x2y)4(2xy2)3Use the Product to a Power Property.(34x8y4)(23x3y6)Simplify.(81x8y4)(8x3y6)Use the Commutative Property.818x8x3y4y6Multiply the constants and add the exponents.648x11y10
Try It 6.2.29

Simplify:

  1. (5n)2(3n10)
  2. (c4d2)5(3cd5)4
Answer
  1. 75n12
  2. 81c24d30
Try It 6.2.30

Simplify:

  1. (a3b2)6(4ab3)4
  2. (2x)3(5x7)
Answer
  1. 256a22b24
  2. 40x10

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

Example 6.2.31

Multiply: (3x2)(4x3)

Solution

(3x2)(4x3)Use the Commutative Property to rearrange the terms.3(4)x2x3Multiply.12x5

Try It 6.2.32

Multiply: (5y7)(7y4)

Answer

35y11

Try It 6.2.33

Multiply: (6b4)(9b5)

Answer

54b9

Example 6.2.34

Multiply: (56x3y)(12xy2)

Solution

(56x3y)(12xy2)Use the Commutative Property to rearrange the terms.5612x3xyy2Multiply.10x4y3

Try It 6.2.35

Multiply: (25a4b3)(15ab3)

Answer

6a5b6

Try It 6.2.36

Multiply: (23r5s)(12r6s7)

Answer

8r11s8

Note

Access these online resources for additional instruction and practice with using multiplication properties of exponents:

  • Multiplication Properties of Exponents

Key Concepts

  • 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 powder 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.
  • Properties of Exponents
    • If a and b are real numbers and m and n are whole numbers, then

Product Property aman=am+nPower Property (am)n=amnProduct to a Power (ab)m=ambm


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