1. 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

2. 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

3. 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

4. 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

5. 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

6. 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.

7. 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.

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

9. 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

10. Power of a Power Rule for Exponents
Power of a Power 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

11. Using Power of a Power Rule
EXAMPLE Using Power of a Power Rule
Use power of a power rule 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

12. Power of a Product Rule 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

13. Using Power of a Product Rule
EXAMPLE Using Power of a Product Rule
Use power of a product rule 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

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

15. Power of a Quotient Rule 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

16. Using Power of a Quotient Rule
EXAMPLE Using Power of a Quotient Rule
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 =

17. If a is any nonzero number, then, a 0 = 1. Example: 25 0 = 1
Zero Exponent
Zero Exponent
If a is any nonzero number, then, a 0 = 1.
Example: = 1 17

18. EXAMPLE 1 Using Zero Exponents
Evaluate each exponential expression.
( a )
= 1
( b ) ( – 7 ) 0
= 1
( c ) – 7 0
= – ( 1 ) = – 1
( d ) g 0
= 1, if g ≠ 0
( e ) 5n 0
= 5 ( 1 ) = 5, if n ≠ 0
( f ) ( 9v ) 0
= 1, if v ≠ 0 18

19. Notice the difference between parts (b) and (c) from Example 1.
Zero Exponents
CAUTION
Notice the difference between parts (b) and (c) from Example 1.
In Example 1 (b) the base is – 7 and in Example 1 (c) the base is 7.
( b ) ( – 7 ) 0
= 1
The base is – 7.
( c ) – 7 0
= – ( 1 ) = – 1
The base is 7. 19

20. If a is any nonzero real number and n is any integer, then
Negative Exponents
Negative Exponents
If a is any nonzero real number and n is any integer, then
Example: 1
a n
a – n = 1
7 2
7 – 2 = 20

21. Using Negative Exponents
EXAMPLE Using Negative Exponents
Simplify by writing each expression with positive exponents. Then
evaluate the expression. 1
8 2 = 1 64 =
( a )
8 –2 1
5 1 = 1 5 =
( b )
5 –1 1
n 8 =
( c )
n –8
when n ≠ 0 21

22. Using Negative Exponents
EXAMPLE Using Negative Exponents
Simplify by writing each expression with positive exponents. Then
evaluate the expression. 1
3 1 =
2 1 +
( d )
3 – –1
Apply the exponents first. 1 3 = 2 + 2 6 = 3 +
Get a common denominator. 5 6 =
Add. 22

23. Negative Exponent Example
CAUTION
A negative exponent does not indicate a negative number; negative
exponents lead to reciprocals.
Expression
Example 1
7 2
7 – 2 = 49
a – n
Not negative 23

24. Quotient Rule for Exponents
If a is any nonzero real number and m and n are any integers, then
(Keep the base and subtract the exponents.)
Example:
a m
a n
a m – n =
3 8
3 2
3 8 – 2 =
3 6 = 24

25. A common error is to write .
CAUTION
3 8
3 2
1 8 – 2 =
1 6
A common error is to write
When using the rule, the quotient should have the same base.
The base here is 3.
If you’re not sure, use the definition of an exponent to write out the
factors.
3 8
3 2
3 8 – 2 =
3 6 1 1
3 8
3 2 =
3 • 3 • 3 • 3 • 3 • 3 • 3 • 3
3 • 3
3 6 = 1 1 1 25

26. Using the Quotient Rule for Exponents
EXAMPLE Using the Quotient Rule for Exponents
Simplify using the quotient rule for exponents. Write answers with
positive exponents.
4 7
4 2
( a )
4 7 – 2 =
4 5
2 3
2 9 = 1
2 6
( b )
2 3 – 9 =
2 –6
9 –3
9 –6
( c )
9 –3 – (–6) =
9 3 26

27. Using the Quotient Rule for Exponents
EXAMPLE Using the Quotient Rule for Exponents
Simplify using the quotient rule for exponents. Write answers with
positive exponents.
x 8
x –2
( d )
x 8 – (–2) =
x 10
when x ≠ 0
n –7
n –4 = 1
n 3
( e )
n –7 – (–4) =
n –3
when n ≠ 0
r –1
r 5 1
r 6 =
( f )
r –1 – 5 =
r –6
when r ≠ 0 27

28. Using the Product Rule with Negative Exponents
EXAMPLE Using the Product Rule with Negative Exponents
Simplify each expression. Assume all variables represent nonzero real
numbers. Write answers with positive exponents.
( a )
5 8 (5 –2 )
(–2) =
5 6 = = 1
6 7
( b )
(6 –1 )(6 –6 )
6 (–1) + (–6) =
6 –7 = = 1
g 2
( c )
g –4 • g 7 • g –5
g (–4) (–5) =
g –2 = 28

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

30. Definitions and Rules for Exponents
If m and n are positive integers and when a ≠ 0, then
Zero Exponent a 0 = (–5) 0 = 1
Negative Exponent
Quotient Exponent
Examples 1
a n
a – n = 1
4 2
4 – 2 =
a m
a n
a m – n =
2 3
2 8
2 3 – 8 =
2 –5 = 1
2 5 = 30

31. Simplifying Expressions vs Evaluating Expressions
NOTE
Make sure you understand the difference between simplifying expressions and evaluating them.
Example:
Simplifying
Evaluating
2 3
2 8 1
2 5 = 1 32 =
2 3 – 8 =
2 –5 = 31