chapter 5 section 2. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. use 0 as an...

Chapter 5 Section 2

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Use 0 as an exponent.

Use negative numbers as exponents.

Use the quotient rule for exponents.

Use combinations of rules.

Integer Exponents and the Quotient Rule5.2

2

3

4


Integer Exponents and the Quotient Rule

In all earlier work, exponents were positive integers. Now, to develop a meaning for exponents that are not positive integers, consider the following list. 4

3

2

2 16

2 8

2 4

Each time the exponent is reduced by 1, the value is divided by 2 (the bases). Using this pattern, the list can be continued to smaller and smaller integers.

12 2 02 1 1 12

2 2 1

24

From the preceding list, it appears that we should define 20 as 1 and negative exponents as reciprocals.

Slide 5.2-3


Objective 1


Slide 5.2-4


so that the product rule is satisfied. Check that the power rules are also valid for a 0 exponent. Thus we define a 0 exponent as follows.


The definitions of 0 and negative exponents must satisfy the rules for exponents from Section 5.1. For example, if 60 = 1, then

and 0 2 2 26 6 1 6 6

Zero Exponent

For any nonzero real number a, a0 = 1.

Example: 170 = 1

0 2 0 2 26 6 6 6

Slide 5.2-5


07

Solution:

1

1 1

Evaluate.

1

07

0 013 2

Slide 5.2-6

EXAMPLE 1 Using Zero Exponents

(1)07 1

2


Objective 2


Slide 5.2-7


Since and we can deduce that 2−n should equal

Is the product rule valid in such a case? For example,


2 12

4 3 1

2 ,8

1

.2n

2 2 2 2 06 6 6 6 1

The expression 6−2 behaves as if it were the reciprocal of 62 since their

product is 1. The reciprocal of 62 is also leading us to define 6−2 as2

1,

62

1.

6

Negative Exponents

For any nonzero real number a and any integer n,

Example:

1.n

na

a

22

3 .1

3

Slide 5.2-8


24

Solution:

2

1

4

34

Simplify.

5 2

5

1

2 2

1

5

3

1

m

5 2

10 10

7

102

2

5

3

31

4

23

5

1 12 5

3 0m m

1

16

64

25

9

Slide 5.2-9

EXAMPLE 2 Using Negative Exponents


Consider the following:

3 4 43

4 3 4 3 3

4

12 1 1 1 3 32

13 2 3 2 1 23

.

Therefore,3 4

4 33.

2 3

2

Changing from Negative to Positive ExponentsFor any nonzero numbers a and b and any integers m and n,

and

Example: and

m n

n m

a b

b a

-m ma b

=b a

5 4

4 5

3 2

2 3

3 3

4 5

5 4

Slide 5.2-10

Use negative numbers as exponents. (cont’d)


Solution:3

2

3

5

Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.

27

25

2

5

4m

h k

3

3

3

2

2y

x

2

3

5

3

5

2

4h

m k

32

32

x

y

9

6

8y

x

We cannot use this rule to change negative exponents to positive exponents if the exponents occur in a sum or difference of terms. For example,

would be written with positive exponents as

2 1

3

5 3

7 2

2

3

1 15 3 .

17

2

Slide 5.2-11

EXAMPLE 3 Changing from Negative to Positive Exponents


Objective 3


Slide 5.2-12


We know that


Notice that the difference between the exponents, 5 − 3 = 2, this is the exponent in the quotient. This example suggests the quotient rule for exponents.

52

3

6 6 6 6 6 66

6 6 6 6.

Quotient Rule for ExponentsFor any nonzero real number a and any integer m and n,

(Keep the same base; subtract the exponents.)

Example:

.m

m nn

aa

a

88 4 4

4

55 5

5

Slide 5.2-13


6

12

x

x

Solution:

7

5

4

4

Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.

5

7

4

4

7 54

4 9 3

5 10 2

8

8

m n

m n

6 ( 12)x 6x24

2

1

45 74 24

1

16

1 1 58

1

m n

4 5 9 10 3 28 m n

16

5

1

8mn

Slide 5.2-14

EXAMPLE 4 Using the Quotient Rule


The product, quotient, and power rules are the same for positive and negative exponents.

Slide 5.2-15



Objective 4

Use combinations of rules.

Slide 5.2-16


Solution:

Simplify. Assume that all variables represent nonzero real numbers.

225

6

y

24

3

3

3

224 4x x

29 2

3 4

3

3

x y

x y

8

3

3

3 8 33 53

2 2 24 4x x 1 2 2 24 x 3 44 x

464x

2 4

2

5

6

y

2

2 4

6

5 y 4

36

25y

9 3 4 2

4

3 x y

x y

6

3

3

y 3

729

y

243

Slide 5.2-17

EXAMPLE 5 Using Combinations of Rules

chapter 5 section 2. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. use 0 as an...

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