Page 736 - Basic College Mathematics with Early Integers

Objectives B.2 MULTIPLICATION PROPERTIES OF EXPONENTS

Use the Product Rule
for Exponents.
Objective Using the Product Rule
Use the Power Rule for
Exponents. Recall from Section 8.1 that an exponent has the same meaning whether the base is
a number or a variable. For example,
Use the Power of a Product Rule 3
# #
3
# #
for Exponents. 5 = 5 5 5 and x = x x x
3 3
3 factors of 5 3 factors of x
We can use this definition of an exponent to discover properties that will help us to
simplify products and powers of exponential expressions.
For example, let’s use the definition of an exponent to find the product of x 3
4
and x .
x 3 # x = 1x x x21x x x x2
4
# #
# # #
# # # # # #
= x x x x x x x
f
7 factors of x
= x 7
Notice that the result is the same if we add the exponents.
4
x 3 # x = x 3+4 = x 7
This suggests the following product rule or property for exponents.

Product Property for Exponents

If m and n are positive integers and a is a real number, then
n
a m # a = a m+n

In other words, to multiply two exponential expressions with the same base,
keep the base and add the exponents.

PRACTICE 1 Example 1 Multiply: y 7 # y 2
Multiply: z 4 # z 8
Solution:
2
y 7 # y = y 7+2 Use the product property for exponents.
= y 9 Simplify.

Work Practice 1

PRACTICE 2 Example 2 Multiply: 3x 5 # 6x 3

Multiply: 7y 5 # 4y 9
Solution:
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#
3x 5 # 6x = 13 621x 5 # x 2 Apply the commutative and associative properties.
3
3
= 18x 5+3 Use the product property for exponents.
= 18x 8
Simplify.
Answers
1. z 12 2. 28y 14 Work Practice 2

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