Page 738 - Basic College Mathematics with Early Integers

714 A PPEND I X B I EXPONENTS AND POLYNOMIALS

In other words, to raise a power to a power, keep the base and multiply the
exponents.

Take a moment to make sure that you understand when to apply
the product rule and when to apply the power rule.

Product Property : Add Exponents Power Property : Multiply Exponents
#
7
x 5 # x = x 5+7 = x 12 1x 2 = x 5 7 = x 35
5 7
#
y 6 # y = y 6+2 = y 8 1y 2 = y 6 2 = y 12
6 2
2
8 2
PRACTICE 5 Example 5 Simplify: 1y 2
3 10
Simplify: 1z 2
Solution:
#
8 2
1y 2 = y 8 2 Use the power property.
= y 16
Work Practice 5

2 9
3 4
PRACTICE 6 Example 6 Simplify: 1a 2 # 1a 2
3 7
4 5
Simplify: 1z 2 # 1z 2
Solution:
3 4 2 9 12 # 18
1a 2 # 1a 2 = a a Use the power property.
= a 12+18 Use the product property.
= a 30 Simplify.
Work Practice 6

Objective Using the Power of a Product Rule

Next, let’s simplify the power of a product.

#
3
1xy2 = xy xy xy Apply the definition of an exponent.
#
# #
# #
= 1x x x21y y y2 Group like bases.
= x y Simplify.
3 3
Notice that the power of a product can be written as the product of powers. This
leads to the following power of a product rule or property.

Copyright 2012 Pearson Education, Inc.
Power of a Product Property for Exponents
If n is a positive integer and a and b are real numbers, then
n n
n
1ab2 = a b

Answers In other words, to raise a product to a power, raise each factor to the power.
5. z 30 6. z 41