Page 737 - Basic College Mathematics with Early Integers

A PPEND I X B . 2 I MULTIPLICATION PROPERTIES OF EXPONENTS 713

Example 3 Multiply: 1-2a b 219a b 2 PRACTICE 3
4 10
5 3
2 5
6 2
Multiply: (-7r s )(-3r s )
Solution: Use properties of multiplication to group numbers and like variables
together.
3
1-2a b 219a b 2 = 1-2 921a 4 # a 21b 10 # b 2
5
4 10
5 3
#
=-18a 4+5 10+3
b
=-18a b
9 13
Work Practice 3
#
Example 4 Multiply: 2x 3 # 3x 5x 6 PRACTICE 4
Multiply: 9y 4 # 3y 2 # y. (Recall
1
Solution: First notice the factor 3x. Since there is one factor of x in 3x, it can also that y = y . )
1
be written as 3x .
# #
x
1
6
6
2x 3 # 3x 1 # 5x = 12 3 521x 3 # # x 2
= 30x 10 Don’t forget that
if an exponent is not written, it
Work Practice 4 is assumed to be 1.

These examples will remind you of the difference between adding
and multiplying terms.
Addition
3
5x + 3x = 15 + 32x = 8x 3
3
3
2
7x + 4x = 7x + 4x 2
Multiplication
3
3
# # #
3
15x 213x 2 = 5 3 x 3 x = 15x 3+3 = 15x 6
2
# # #
2
17x214x 2 = 7 4 x x = 28x 1+2 = 28x 3
Objective Using the Power Rule
Next suppose that we want to simplify an exponential expression raised to a power.
2 3
To see how we simplify 1x 2 , we again use the definition of an exponent.
2 3 2 # 2 # 2 Apply the definition of an exponent.
1x 2 = 1x 2 1x 2 1x 2
e
3 factors for x 2
= x 2+2+2 Use the product property for exponents
= x 6 Simplify.

Notice the result is exactly the same if we multiply the exponents.
#
2 3
1x 2 = x 2 3 = x 6
This suggests the following power rule or property for exponents.

Power Property for Exponents

If m and n are positive integers and a is a real number, then
m n
1a 2 = a m # n Answers
8 7
3. 21r s 4. 27y 7