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Columbus State Community College

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Integer Exponents and the Quotient Rule

- Use 0 as an exponent.
- Use negative numbers as exponents.
- Use the quotient rule for exponents.
- Use the product rule with negative exponents.

Using Zero Exponents

EXAMPLE 1 Using Zero Exponents

Evaluate each exponential expression.

( a ) 310

= 1

( b ) ( –7 )0

= 1

( c ) –70

= – ( 1 ) = – 1

( d ) g0

= 1,if g ≠ 0

( e ) 5n0

= 5 ( 1 ) = 5, if n≠ 0

( f ) ( 9v )0

= 1, if v≠ 0

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 ) –70

= – ( 1 ) = – 1

The base is 7.

7 – 2

=

72

1

a– n

=

an

Negative ExponentsNegative Exponents

If a is any nonzero real number and n is any integer, then

Example:

1

1

1

1

=

=

=

=

=

( c )

( b )

( a )

5 –1

8 –2

n–8

51

82

5

n8

64

Using Negative ExponentsEXAMPLE 2 Using Negative Exponents

Simplify by writing each expression with positive exponents. Then

evaluate the expression.

when n ≠ 0

3

2

5

1

1

1

+

=

=

=

+

+

=

( d )

3 –1 + 2 –1

2

6

6

6

21

3

31

Using Negative ExponentsEXAMPLE 2 Using Negative Exponents

Simplify by writing each expression with positive exponents. Then

evaluate the expression.

Apply the exponents first.

Get a common denominator.

Add.

1

7 – 2

=

=

72

49

Negative ExponentCAUTION

A negative exponent does not indicate a negative number; negative

exponents lead to reciprocals.

Expression

Example

a– n

Not negative

am

=

am – n

an

38

=

38 – 2

=

36

32

Quotient Rule for ExponentsQuotient 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:

38

38

38

3 • 3 • 3 • 3 • 3 • 3 • 3 • 3

36

=

=

18 – 2

38 – 2

=

=

16

36

=

=

=

36

32

32

32

3 • 3

1

Common ErrorCAUTION

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.

1

1

1

1

23

9–3

47

=

=

23 – 9

47 – 2

=

=

45

2–6

26

29

9–6

42

=

=

9–3 – (–6)

=

93

Using the Quotient Rule for ExponentsEXAMPLE 3 Using the Quotient Rule for Exponents

Simplify using the quotient rule for exponents. Write answers with

positive exponents.

( a )

( b )

( c )

n–7

r–1

1

1

x8

=

=

x8 – (–2)

n–7 – (–4)

=

=

x10

n–3

r6

x–2

n3

r5

n–4

=

=

r–1 – 5

=

=

r–6

Using the Quotient Rule for ExponentsEXAMPLE 3 Using the Quotient Rule for Exponents

Simplify using the quotient rule for exponents. Write answers with

positive exponents.

( d )

when x≠ 0

( e )

when n≠ 0

when r≠ 0

( f )

1

58 (5–2)

( a )

=

58 + (–2)

=

56

g2

67

=

( b )

=

6(–1) + (–6)

=

6–7

(6–1 )(6–6 )

( c )

=

g–2

g–4• g7• g–5

=

g(–4) + 7 + (–5)

=

Using the Product Rule with Negative ExponentsEXAMPLE 4 Using the Product Rule with Negative Exponents

Simplify each expression. Assume all variables represent nonzero real

numbers. Write answers with positive exponents.

3

32

a

a m

m

4

42

=

=

b

b m

Definitions and Rules for ExponentsDefinitions and Rules for Exponents

If m and n are positive integers, then

Product Ruleam • an = am + n34 • 32 = 34 + 2 = 36

Power Rule (a)( am )n = am n( 35 )2 = 35 • 2 = 310

Power Rule (b)( a b )m = am bm( 5a )8 = 58 a8

Power Rule (c)(b≠ 0 )

Examples

a– n

=

an

1

4 – 2

=

42

am

=

am – n

23

an

=

23 – 8

=

2–5

28

1

=

25

Definitions and Rules for ExponentsDefinitions and Rules for Exponents

If m and n are positive integers and when a ≠ 0, then

Zero Exponenta0 = 1 (–5)0= 1

Negative Exponent

Quotient Exponent

Examples

23

1

1

=

23 – 8

=

2–5

=

=

28

25

32

Simplifying Expressions vs Evaluating ExpressionsNOTE

Make sure you understand the difference between simplifying expressions and evaluating them.

Example:

Simplifying

Evaluating

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