Chapter 8 Section 7. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Using Rational Numbers as Exponents Define and use expressions.

Chapter 8 Section 7

Objectives

1

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

Using Rational Numbers as Exponents

Define and use expressions of the form a1/n.

Define and use expressions of the form am/n.

Apply the rules for exponents using rational exponents.

Use rational exponents to simplify radicals.

8.7

2

3

4


Objective 1


Slide 8.7-3



Now consider how an expression such as 51/2 should be defined, so that all the rules for exponents developed earlier still apply. We define 51/2 so that

51/2 · 51/2 = 51/2 + 1/2 = 51 = 5.

This agrees with the product rule for exponents from Section 5.1. By definition,

Since both 51/2 · 51/2 and equal 5,

this would seem to suggest that 51/2 should equal

Similarly, then 51/3 should equal

5 5 5.

5 5

3 5.5.

Review the basic rules for exponents:

m n m na a a m

m nn

aa

a nm mna a

Slide 8.7-4


a1/nIf a is a nonnegative number and n is a positive integer, then

1/ .n na a

Slide 8.7-5


Notice that the denominator of the rational exponent is the index of the radical.


Simplify.

491/2

10001/3

811/4

Solution:

49 7

3 1000 10

4 81 3

Slide 8.7-6

EXAMPLE 1 Using the Definition of a1/n


Objective 2


Slide 8.7-7



Now we can define a more general exponential expression, such as 163/4. By the power rule, (am)n = amn, so

333/ 4 1/ 4 3416 16 16 2 8.

However, 163/4 can also be written as

Either way, the answer is the same. Taking the root first involves smaller numbers and is often easier. This example suggests the following definition for a m/n.

1/ 4 1/ 43/ 4 3 416 16 4096 4096 8.

am/nIf a is a nonnegative number and m and n are integers with n > 0, then

/ 1/ .mmm n n na a a

Slide 8.7-8


Evaluate.

95/2

85/3

–272/3

Solution:

51/ 29 53

51/38 52

21/327 9

243

32

23

Slide 8.7-9

EXAMPLE 2 Using the Definition of am/n


Earlier, a–n was defined as

for nonzero numbers a and integers n. This same result applies to negative rational exponents.

Using the definition of am/n.

1nn

aa

a−m/nIf a is a positive number and m and n are integers, with n > 0, then

//

1.m n

m na

a

A common mistake is to write 27–4/3 as –273/4. This is incorrect. The negative exponent does not indicate a negative number. Also, the negative exponent indicates to use the reciprocal of the base, not the reciprocal of the exponent.

Slide 8.7-10


Solution:

Evaluate.

36–3/2

81–3/4

3/ 2

1

36

3/ 4

1

81

31/ 2

1

36

3

1

6

1

216

31/ 4

1

81

3

1

3

1

27

Slide 8.7-11

EXAMPLE 3 Using the Definition of a−m/n


Objective 3


Slide 8.7-12



All the rules for exponents given earlier still hold when the exponents are fractions.

Slide 8.7-13


Solution:

Simplify. Write each answer in exponential form with only positive exponents.

1/3 2/37 72/3

1/3

9

9

5/327

8

1/ 2 2

5/ 2

3 3

3

1/3 2 /37 7

2/3 1/39 9

5/3

5/3

27

8

51/3

51/3

27

8

5

5

3

2

1/ 2 4/ 2 5/ 23 2/ 23 3

Slide 8.7-14

EXAMPLE 4 Using the Rules for Exponents with Fractional Exponents


Simplify. Write each answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.

Solution:

62/3 1/3 2a b c

2/3 1/3

1

r r

r

32/3

1/ 4

a

b

6 6 62/3 1/3 2a b c 12/3 6/3 12a b c 4 2 12a b c

2/3 1/3 3/3r 6/3r 2r

32/3

31/ 4

a

b

6/3

3/ 4

a

b

2

3/ 4

a

b

Slide 8.7-15

EXAMPLE 5 Using Fractional Exponents with Variables


Objective 4


Slide 8.7-16



Sometimes it is easier to simplify a radical by first writing it in exponential form.

Slide 8.7-17


Simplify each radical by first writing it in exponential form.

4 212

36 x

1/ 4212 1/ 212 12

1/ 63x 1/ 2x 0x x

Solution:

2 3

Slide 8.7-18

EXAMPLE 6 Simplifying Radicals by Using Rational Exponents

Chapter 8 Section 7. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Using Rational Numbers as Exponents Define and use expressions.

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