1. 13
Exponents and Polynomials

2. 13.2 The Product Rule and Power Rules for Exponents
Objectives
1. Use exponents. 2. Use the product rule for exponents. 3. Use the rule (am)n = amn. 4. Use the rule (ab)m = ambm. 5. Use the rule (a/b)m = am/bm. 6. Use combinations of the rules for exponents. 7. Use the rules for exponents in a geometry application.

3. Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate.
Use Exponents
Example 1
Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate.
Since 2 occurs as a factor 6 times, the base is 2 and the exponent is 6.
The exponential expression is 26, read “2 to the sixth power” or simply “2 to the sixth.”
Exponent (or Power)
2 = 2 · 2 · 2 · 2 · 2 · 2 6
= 64
Base
6 factors of 2

4. Example 2 Evaluate. Name the base and the exponent. Base Exponent
Use Exponents
Example 2 Evaluate. Name the base and the exponent.
Base
Exponent
(a) = 2 · 2 · 2 · 2 4 2 4
= 16
(b) – 2 4 4
= –1 · 2
= –1 · 2 · 2 · 2 · 2 2 4
= –16
(c) (–2) 4
= (–2) (–2) (–2) (–2) –2 4
= 16

5. In summary, and are not necessarily the same.
Use Exponents
CAUTION
Expression
Base
Exponent
Example
– a n
– 5 2 a n
= – ( 5 · 5 ) = – 25
(–a) n
(– 5) 2
(–a) n
= (–5) (–5) = 25
– a n
(–a) n
In summary, and are not necessarily the same.

6. Use the Product Rule for Exponentsa m
· a n
= a
m + n
For any positive integers m and n,
(Keep the same base and add the exponents.) 5 3 · 4
Example: =
3 + 4 = 7
( 5 · 5 · 5 )
( 5 · 5 · 5 · 5 )

7. Use the Product Rule for Exponents
CAUTION
Do not multiply the bases when using the product rule. Keep the same base and add the exponents.
Example:

8. Use the Product Rule for Exponents
Example 3
Use the product rule for exponents to simplify, if possible. 5 8
(a) 7 · 7
= 7
5 + 8
= 7 13 4 5
(b) (–2) (–2)
= (– 2)
4 + 5
= (– 2) 9 5
(c) y · y 5
= y · y 1
= y
1 + 5
= y 6

9. Use the Product Rule for Exponents
Example 3 (concluded)
Use the product rule for exponents to simplify, if possible. 2 4
(d) n · n
= n
2 + 4
= n 6 2
(e) 3 · 2
The product rule does not apply because the bases are different.
= · = 36 3 2
(f)
The product rule does not apply because it is a sum, not a product.
= = 12

10. Use the Product Rule of Exponents
Example 4
Commutative and associative properties
Multiply; product rule
Add the exponents.

11. Use the Product Rule for Exponents
CAUTION
Be sure you understand the difference between adding and multiplying exponential expressions. For example, 2
7k k 2
= ( )k = 10k ,
Add. 2
7k k 4
2 + 2
= ( 7 · 3 )k = 21k .
Multiply.
but,

12. Power Rule (a) for Exponents For any positive integers m and n,
Use the Rule (am)n = amn
Power Rule (a) for Exponents
For any positive integers m and n, m n
( a )
m n
= a .
(Raise a power to a power by multiplying exponents.) 3 2
( 4 )
3 · 2
= 4 6 =
Example:

13. Use power rule (a) for exponents to simplify.
Use the Rule (am)n = amn
Example 5
Use power rule (a) for exponents to simplify.
(a) ( 3 ) 2 5
= 3
2 · 5
= 3 10
(b) ( 4 ) 8 6
= 4
8 · 6
= 4 48
(c) ( n ) 7 3
= n
7 · 3
= n 21
(d) ( 2 ) 5 8
= 2
5 · 8
= 2 40

14. Use the Rule (ab)m = ambm
Power Rule (b) for Exponents
For any positive integer m, m
( ab )
m m
= a b .
(Raise a product to a power by raising each factor to the power.) 3
( 5h ) 3
= h .
Example:

15. Use the Rule (ab)m = ambm
Example 6
Use power rule (b) for exponents to simplify.
(a) ( 2abc ) 4
= a b c 4
Power rule (b)
= 16 a b c 4
(b) 5 ( x y ) 2 3
= 5 ( x y ) 2 6
Power rule (b)
= 5x y 2 6

16. Use the Rule (ab)m = ambm
Example 6 (continued)
Use power rule (b) for exponents to simplify.
(c) 7 ( 2m n p ) 7 5 3
= 7 [ 2 ( m ) ( n ) ( p ) ] 3 5 7 1
Power rule (b)
= 7 [ 8 m n p ] 3 15 21
Power rule (a)
= m n p 3 15 21

17. Use the Rule (ab)m = ambm
Example 6 (concluded)
Use power rule (b) for exponents to simplify.
(d) ( –3 ) 5 4
= ( – 1 · 3 ) 5 4
– a = – 1 · a
= ( – 1 ) ( 3 ) 5 4
Power rule (b)
= – 1 · 3 20
Power rule (a)
= – 3 20

18. Use the Rule (ab)m = ambm
CAUTION
Power rule (b) does not apply to a sum:

19. Use the Rule (a/b)m = am/bm
Power Rule (c) for Exponents
For any positive integer m, m a b =
( b ≠ 0 ).
(Raise a quotient to a power by raising both the numerator and the denominator to the power.)
Example: 2 3 4 =

20. Use the Rule (a/b)m = am/bm
Example 7
Use power rule (c) for exponents to simplify. 4 2 5 4 2 5 = 16
625 =
(a) 7 x y 7 x y =
(b)
y ≠ 0

21. For any positive integers m and n: Examples
Rules for Exponents
For any positive integers m and n:
Examples a m
· a n
= a
m + n 3 4
· 3 5
= 3 9
Product rule m n
( a )
m n
= a
(a) 4 5
( 2 ) 20
= 2
Power rules m
m m
(b)
( ab )
= a b 3
( 4k )
= 4 k
(c) m a b = 3 4 7 =
( b ≠ 0 ).

22. Use Combinations of the Rules for Exponents
Example 8
Simplify each expression.
(a)
· 3 5 2 3 4
= · 2 3 4 5 1
Power rule (c) = 2 · · 5 3 1
Multiply fractions. = 3 4 2 7
Product rule

23. Use Combinations of the Rules for Exponents
Example 8 (continued)
Simplify each expression.
(b)
( 7m n ) ( 7m n ) 2 3 5 =
( 7m n ) 2 8
Product rule =
7 m n 16 8
Power rule (b)

24. Use Combinations of the Rules for Exponents
Example 8 (continued)
Simplify each expression.
(c)
( 2x y ) ( 2 x y ) 10 6 3 7 4 =
( 2 ) ( x ) ( y ) · ( 2 ) ( x ) ( y ) 10 7 4 6 3
Power rule (b) =
2 · x · y · · x · y 10 70 40 6 18
Power rule (a) =
2 · · x · x · y · y 70 6 40 10 18
Commutative and associative properties =
2 · x · y 16 88 46
Product rule

25. Use Combinations of the Rules for Exponents
Example 8 (concluded)
Simplify each expression.
(d)
( – g h ) ( – g h ) 3 2 5 =
( – 1 g h ) ( – 1 g h ) 3 2 5 =
( – 1 ) ( g ) ( h ) ( – 1 ) ( g ) ( h ) 2 6 3 15
Power rule (b) =
( – 1 ) ( g ) ( h ) 5 12 17
Product rule =
– 1 g h 12 17

26. Use the Rules for Exponents in a Geometry Application
Example 9
Find a polynomial that represents the area of the geometric figure. 4x 3 2x 2
(a)
Use the formula for the area of a rectangle, A = LW.
A = ( 4x )( 2x ) 3 2
A = 8x 5
Product rule

27. Use the Rules for Exponents in a Geometry Application
Example 9
Find a polynomial that represents the area of the geometric figure. 8n 5 3n 4 1 2
(b)
Use the formula for the area of a triangle, A = LW.
A = ( 3n ) ( 8n ) 4 5 1 2
A = ( 24n ) 9 1 2
Product rule
A = 12n 9