1. Columbus State Community College
Chapter 8 Section 1
The Product Rule and Power Rules for Exponents

2. The Product Rule and Power Rules for Exponents
Review the use of exponents.
Use the product rule for exponents.
Use the exponent rule ( a m ) n = a m n.
Use the exponent rule ( a b ) m = a m b m.
Use the exponent rule = a b m
a m
b m

3. Review of Using Exponents
EXAMPLE Review of Using Exponents
Write 5 • 5 • 5 • 5 in exponential form, and find the value of the
exponential expression.
Since 5 appears as a factor 4 times, the base is 5 and the
exponent is 4.
Writing in exponential form, we have 5 4.
5 4 = 5 • 5 • 5 • 5 = 625

4. Evaluating Exponential Expressions
EXAMPLE Evaluating Exponential Expressions
Evaluate each exponential expression. Name the base and the
exponent.
Base
Exponent
( a ) 2 4
= 2 • 2 • 2 • 2 = 16 2 4
( b ) – 2 4
= – ( 2 • 2 • 2 • 2 ) = – 16 2 4
( c ) ( – 2 ) 4
= ( – 2 )( – 2 )( – 2 )( – 2 ) = 16
– 2 4

5. Understanding the Base
CAUTION
It is important to understand the difference between parts (b) and (c)
of Example 2. In – 2 4 the lack of parentheses shows that the
exponent 4 applies only to the base 2.
In ( – 2 ) 4 the parentheses
show that the exponent 4 applies to the base – 2.
In summary,
– a m and ( – a ) m mean different things. The exponent applies
only to what is immediately to the left of it.
Expression
Base
Exponent
Example
– a n a n
– 5 2 = – ( 5 • 5 ) = – 25
( – a ) n
– a n
( – 5 ) 2 = ( – 5 ) ( – 5 ) = 25

6. Product Rule for Exponents
If m and n are positive integers, then a m • a n = a m + n (Keep the same base and add the exponents.)
Example: • = = 3 6

7. Common Error Using the Product Rule
CAUTION
Avoid the common error of multiplying the bases when using the
product rule.
Keep the same base and add the exponents.
3 4 • ≠ 9 6
3 4 • = 3 6

8. Using the Product Rule
EXAMPLE Using the Product Rule
Use the product rule for exponents to find each product, if possible.
( a ) • 6 7
= = by the product rule.
( b ) ( – 7 ) 1 ( – 7 ) 5
( b ) ( – 7 ) 1 ( – 7 ) 5
= ( – 7 ) = ( – 7 ) by the product rule.
( c ) • 3 2
The product rule doesn’t apply. The bases are different.
( d ) x 9 • x 5
= x = x by the product rule.

9. Using the Product Rule
EXAMPLE Using the Product Rule
Use the product rule for exponents to find each product, if possible.
( e )
The product rule doesn’t apply because this is a sum.
( f ) ( 5 m n 4 ) ( – 8 m 6 n 11 )
= ( 5 • – 8 ) • ( m m 6 ) • ( n 4 n 11 ) using the commutative and
associative properties.
= – 40 m 7 n by the product rule.

10. Product Rule and Bases
CAUTION
The bases must be the same before we can apply the product rule for
exponents.

11. Understanding Differences in Exponential Expressions
CAUTION
Be sure you understand the difference between adding and
multiplying exponential expressions. Here is a comparison.
Adding expressions 3 x x 4 = 5 x 4
Multiplying expressions ( 3 x 4 ) ( 2 x 5 ) = 6 x 9

12. Power Rule (a) for Exponents
If m and n are positive integers, then ( a m ) n = a m n (Raise a power to a power by multiplying exponents.)
Example: ( 3 5 ) 2 = 3 5 • 2 = 3 10

13. Using Power Rule (a)
EXAMPLE Using Power Rule (a)
Use power rule (a) to simplify each expression. Write answers in
exponential form.
( a ) ( 3 2 ) 7
= • 7 =
( b ) ( 6 5 ) 9
= • 9 =
( c ) ( w 4 ) 2
= w 4 • 2 = w 8

14. Power Rule (b) for Exponents
If m is a positive integer, then ( a b ) m = a m b m (Raise a product to a power by raising each factor to the power.)
Example: ( 5a ) 8 = 5 8 a 8

15. Using Power Rule (b)
EXAMPLE Using Power Rule (b)
Use power rule (b) to simplify each expression.
( a ) ( 4n ) 7
= n 7
( b ) 2 ( x 9 y 4 ) 5
= 2 ( x 45 y 20 ) = 2 x 45 y 20
( c ) 3 ( 2 a 3 b c 4 ) 2
= 3 ( 2 2 a 6 b 2 c 8 )
= 3 ( 4 a 6 b 2 c 8 )
= 12 a 6 b 2 c 8

16. The Power Rule
CAUTION
Power rule (b) does not apply to a sum.
( x ) 2 ≠ x  Error
You will learn how to work with ( x ) 2 in more advanced
mathematics courses.

17. Power Rule (c) for Exponents
If m is a positive integer, then =
(Raise a quotient to a power by raising both the numerator and the
denominator to the power. The denominator cannot be 0.)
Example: = a b m
a m
b m 3 4 2
3 2
4 2

18. 3 2 Using Power Rule (c) EXAMPLE 6 Using Power Rule (c)
Simplify each expression.
( a ) 5 8 3 =
5 3
8 3 =
125
512
( b )
3a 9
7 b c 3 2
( 3a 9 ) 2
( 7 b 1 c 3 ) 2 =
3 2 a 18
7 2 b 2 c 6 =
9 a 18
49 b 2 c 6 =

19. m 2 Rules for Exponents Rules for Exponents
If m and n are positive integers, then
Product Rule a m • a n = a m + n • = = 3 6
Power Rule (a) ( a m ) n = a m n ( 3 5 ) 2 = 3 5 • 2 = 3 10
Power Rule (b) ( a b ) m = a m b m ( 5a ) 8 = 5 8 a 8
Power Rule (c) ( b ≠ 0 )
Examples
a m
b m a b m =
3 2
4 2 3 4 2 =

20. The Product Rule and Power Rules for Exponents
Chapter 8 Section 1 – Completed
Written by John T. Wallace