Properties of Exponents

Section 4.1

Lehmann, Intermediate Algebra, 4edSection 4.1

For any counting number n,

We refer to bn at the power; the nth power of b, or b

raised to the nth power.

We call b the base and n

the exponent.

Slide 2

Definition: ExponentD e f i n i t i o n o f a n E x p o n e n t

Definition

nb b b b b=

N factors of b

Lehmann, Intermediate Algebra, 4edSection 4.1

Two powers of b have specific names. We refer to b2

as the square of b or b squared. We refer to b3 as the

cube of b or b cubed.

For –bn, we compute bn before finding the opposite.

For –24, the base is 2, not –2. If we want the base –2

Slide 3

Definition: ExponentD e f i n i t i o n o f a n E x p o n e n t

Definition

Clarify

Lehmann, Intermediate Algebra, 4edSection 4.1

• Use a graphing calculator to check both

computations

• To find –24, press (–) 2 ^ 3 ENTER

Slide 4

Definition: ExponentD e f i n i t i o n o f a n E x p o n e n t

Calculator

Lehmann, Intermediate Algebra, 4edSection 4.1 Slide 5

Properties of ExponentsP r o p e r t i e s o f E x p o n e n t

Properties

Lehmann, Intermediate Algebra, 4edSection 4.1

Show that b5b3 = b5.

• Writing b5b3 without exponents, we see that

• Use calculator to verify by using various bases and

examining the table

Slide 6

Properties of ExponentsP r o p e r t i e s o f E x p o n e n t

Example

Solution

Lehmann, Intermediate Algebra, 4edSection 4.1

Show that bmbn = bm+n, where m and n are counting

numbers.

Slide 7

Properties of ExponentsP r o p e r t i e s o f E x p o n e n t

Example

Solution Continued

Lehmann, Intermediate Algebra, 4edSection 4.1

• Write bmbn without exponents:

Show that , n is a counting number and

c ≠ 0.

Slide 8

Properties of ExponentsP r o p e r t i e s o f E x p o n e n t

Solution

Examplen n

n

b b

c c

=

Lehmann, Intermediate Algebra, 4edSection 4.1

• Write without exponents:

Slide 9

Properties of ExponentsP r o p e r t i e s o f E x p o n e n t

Solutionn

b

c

Lehmann, Intermediate Algebra, 4edSection 4.1

An expression involving exponents is simplified if

1. It includes no parentheses.

2. Each variable or constant appears as a base as few

times as possible. For example, we write x2x4 = x6

3. Each numerical expression (such as 72) has been

calculated, and each numerical fraction has been

simplified.

4. Each exponent is positive.

Slide 10

Simplifying Expressions Involving ExponentsS i m p l i f y i n g E x p r e s s i o n s I n v o l v i n g E x p o n e n t s

Property

Lehmann, Intermediate Algebra, 4edSection 4.1

Simplify.

Slide 11

Simplifying Expressions Involving ExponentsS i m p l i f y i n g E x p r e s s i o n s I n v o l v i n g E x p o n e n t s

Example

( )

( )( )

52 3

3 4 2 2

7 6

2 5

47 8

2 5 3

1. 2

2. 3 2

33.

12

244.

16

b c

b c b c

b c

b c

b c

b c d

Lehmann, Intermediate Algebra, 4edSection 4.1 Slide 12

Simplifying Expressions Involving ExponentsS i m p l i f y i n g E x p r e s s i o n s I n v o l v i n g E x p o n e n t s

Solution

Lehmann, Intermediate Algebra, 4edSection 4.1 Slide 13

Simplifying Expressions Involving ExponentsS i m p l i f y i n g E x p r e s s i o n s I n v o l v i n g E x p o n e n t s

Solution Continued

Lehmann, Intermediate Algebra, 4edSection 4.1

• 3b2 and (3b)2 are not equivalent

• 3b2 base is b, and (3b)2 base is the 3b

• Typical error looks like

Slide 14

Simplifying Expressions Involving ExponentsS i m p l i f y i n g E x p r e s s i o n s I n v o l v i n g E x p o n e n t s

Warning

Lehmann, Intermediate Algebra, 4edSection 4.1

What is the meaning of b0? The property

is to be true for m = n, then

So, a reasonable definition of b0 is 1.

For b ≠ 0,

b0 = 1

Slide 15

Simplifying Expressions Involving ExponentsZ e r o a s a n E x p o n e n t

Introductionm

m n

n

bb

b

−=

01 , 0n

n n

n

bb b b

b

−= = =

Definition

Lehmann, Intermediate Algebra, 4edSection 4.1

• 70 = 1, (–3)0 = 1, and (ab)0 = 1, where ab ≠ 0

Slide 16

Simplifying Expressions Involving ExponentsZ e r o a s a n E x p o n e n t

Illustration

Lehmann, Intermediate Algebra, 4edSection 4.1

If n is a negative integer, what is the meaning of bn?

What is the meaning of a negative exponent? If the

property is true for m = 0, then

So, we would define b–n to be .

Slide 17

Negative ExponentsN e g a t i v e E x p o n e n t s

Introduction

mm n

n

bb

b

−=

001

, 0n n

n n

bb b b

b b

− −= = =

1nb

Lehmann, Intermediate Algebra, 4edSection 4.1

If b ≠ 0 and n is a counting number, then

In words: To find b–n, take its reciprocal and switch

the sign of the exponent.

For example

Slide 18

Negative Integer ExponentsN e g a t i v e E x p o n e n t s

Definition

1n

nb

b

− =

Illustration

2 5

2 5

1 1 13 and

93b

b

− −= = =

Lehmann, Intermediate Algebra, 4edSection 4.1

We write in another form, where b ≠ 0 and n is a

counting number:

Slide 19

Negative ExponentsN e g a t i v e E x p o n e n t s

Introduction1

nb−

Lehmann, Intermediate Algebra, 4edSection 4.1

If b ≠ 0 and n is a counting number, then

In words: To find , take its reciprocal and switch t

he sign of the exponent.

For example,

Slide 20

Negative ExponentsN e g a t i v e E x p o n e n t s

Definition

1 n

nb

b−=

1nb−

Example4 8

4 8

1 128 16 and .

2b

b− −= = =

Lehmann, Intermediate Algebra, 4edSection 4.1

Simplify.

Slide 21

Simplifying More Expressions Involving ExponentsS i m p l i f y M o r e E x p r e s s i o n s I n v o l v i n g E x p o n e n t s

Example

7 1 1

3

51. 9 2. 3. 3 4b

b

− − −

−+

Solution

Lehmann, Intermediate Algebra, 4edSection 4.1 Slide 22

Properties of Integer ExponentsS i m p l i f y M o r e E x p r e s s i o n s I n v o l v i n g E x p o n e n t s

Properties