copyright © 2015, 2011, 2007 pearson education, inc. 1 1 chapter 8 rational exponents, radicals,...

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1

Chapter 8

Rational Exponents,

Radicals, and Complex Numbers

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2

Rational Exponents, Radicals, and Complex Numbers

8.1 Radical Expressions and Functions8.2 Rational Exponents8.3 Multiplying, Dividing, and Simplifying

Radicals8.4 Adding, Subtracting, and Multiplying

Radical Expressions8.5 Rationalizing Numerators and

Denominators of Radical Expressions8.6 Radical Equations and Problem Solving 8.7 Complex Numbers

CHAPTER

8

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3

Rational Exponents

1. Evaluate rational exponents.2. Write radicals as expressions raised to

rational exponents.3. Simplify expressions with rational number

exponents using the rules of exponents.4. Use rational exponents to simplify radical

expressions.

8.2

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4

Rational exponent: An exponent that is a rational number.

Rational Exponents with a Numerator of 1

a1/n = where n is a natural number other than 1.,n a

Note: If a is negative and n is odd, then the root is negative.If a is negative and n is even, then there is no real number root.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5

Example

Rewrite using radicals, then simplify if possible. a. 491/2 b. 6251/4 c. (216)1/3

Solution

a.

b.

c.

1/ 249

1/4625

1/3216

49 7

4 625 5

3 216 6

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6

continued

Rewrite using radicals, then simplify. d. (16)1/4 e. 491/2 f. y1/6

Solution

d.

e.

f.

1/4( 16)

1/249

1/6y

4 16 There is no real number answer.

49 7

6 y

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7

continued

Rewrite using radicals, then simplify. g. (100x8)1/2 h. 9y1/5 i.

Solution

d.

e.

f.

8 1/2(100 )x

1/59y1/28

49

w

8 4100 10x x

59 y

8 4

49 7

w w

1/28

49

w

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8

General Rule for Rational Exponents

where a 0 and m and n are natural numbers other than 1.

/ ,m

nm n m na a a

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9

Example

Rewrite using radicals, then simplify, if possible. a. 272/3 b. 2433/5 c. 95/2

Solutiona.

b.

c.

2/3 1/3 227 (27 )

3/5 1/5 3243 (243 )

5/2 1/2 59 (9 )

23( 27) 23 9

35( 243) 33 27

5(3) 243 5( 9)

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10

continued

Rewrite using radicals, then simplify, if possible. d. e. f.

Solutiond.

e.

f.

33/21 1

16 16

52/5 2x x

3/5 35(4 1) (4 1)x x

31

4

1

64

3/21

16

2/5x 3/5(4 1)x

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11

Negative Rational Exponents

where a 0, and m and n are natural numbers with n 1.

//

1,m n

m na

a

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12

Example

Rewrite using radicals; then simplify if possible. a. 251/2 b. 272/3

Solutiona.

b.

1/ 21/ 2

125

25

23

1

27

1 1

525

2/32/3

127

27 2

1 1

3 9

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13

continued

Rewrite using radicals; then simplify if possible. c. d.

Solutionc.

1/2

1/2

25 1

36 25

36

2/3

1

( 27)

1

2536

2/3d. ( 27)

23

1

( 27)

1/225

36

156

6

5

2/3( 27)

2

1

( 3)

1

9

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14

Example

Write each of the following in exponential form.

a.

Solution

6 5x

6 5x 5/ 6x

b. 34

1

x

a.

b. 34

1

x

3/4

1

x3/4x

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15

continued

Write each of the following in exponential form.

c.

Solution

45 x

45 x 4/5x

d. 34 5 2x

c.

d. 34 5 2x 3/ 45 2x

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16

Rules of Exponents Summary(Assume that no denominators are 0, that a and b are

real numbers, and that m and n are integers.)Zero as an exponent: a0 = 1, where a 0.

00 is indeterminate.Negative exponents:

Product rule for exponents:Quotient rule for exponents:Raising a power to a power:Raising a product to a power:Raising a quotient to a power:

1 , n

n

aa

m n m na a a

m n m na a a

nm mna a

n n nab a b

n na bb a

n

n

na ab b

1 ,n

n

aa

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17

Example

Use the rules of exponents to simplify. Write the answer with positive exponents.

Solution

3/ 4 1/ 4y y

3/ 4 1/ 4y y 3/ 4 ( 1/ 4)y 2/ 4y1/ 2y

Use the product rule for exponents. (Add the exponents.)

Add the exponents.

Simplify the rational exponent.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18

Example

Use the rules of exponents to simplify. Write the answer with positive exponents.

Solution

1/3 1/63 4a a

1/3 1/63 4a a 1/3 1/612a 2/6 1/612a

3/6 1/212 or 12a a

Use the product rule for exponents. (Add the exponents.)

Rewrite the exponents with a common denominator of 6.

Add the exponents.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 19

Example

Use the rules of exponents to simplify. Write the answer with positive exponents.

Solution

Use the quotient for exponents. (Subtract the exponents.)

Rewrite the subtraction as addition.

Add the exponents.

5/ 6

1/ 6

y

y

5/ 6

1/ 6

y

y 5/ 6 ( 1/ 6)y

5/ 6 1/ 6y

y

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 20

Example

Use the rules of exponents to simplify. Write the answer with positive exponents.

Solution

Add the exponents.

2/5 3/53 5y y

2/5 3/53 5y y 2/5 3/515y

1/515y

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 21

Example

Use the rules of exponents to simplify. Write the answer with positive exponents.

Solution

27/8m

27/8m (7/8)2m7/4m

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 22

Example

Use the rules of exponents to simplify. Write the answer with positive exponents.

Solution

32/5 4/53a b

32/5 4/53a b 3 2/5 3 4/5 33 ( ) ( )a b

(2/5)3 (4/5)327a b

6/5 12/527a b

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 23

Example

Use the rules of exponents to simplify. Write the answer with positive exponents.

Solution

8/3 3

6

(2 )x

x

8/3 3

6

(2 )x

x

3 8/3 3

6

2 ( )x

x

8

6

8x

x

8 68x 28x

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 24

Example

Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values.

a. b.

Solution

4 64

4a. 64 1/4642 1/4(8 )

21/48 1/28

6 10x

8

6 10b. x 10/6x5/3x

3 5x

3 2x x

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 25

continued

Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values.

c.

Solution

6 28 w y

6 28c. w y6 2 1/8( )w y

61/8 21/8w y

3/4 1/4w y

1/43w y

34 w y

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 26

ExamplePerform the indicated operations. Write the result

using a radical.

Solution

b.a. 34x x

a. 34x x 1/ 2 3/ 4x x 1/ 2 3/ 4x 2/ 4 3/ 4x 5/ 4x

54 x

6 7

3

x

x

b.6 7

3

x

x

7 / 6

1/3

x

x

7 / 6 1/3x 7 / 6 2/ 6x

5/ 6x6 5x

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 27

continuedPerform the indicated operations. Write the result

using a radical.

Solution

c. 45 4

c. 45 4 1/2 1/45 4 2/4 1/45 4

1/425 4 1/4(25 4)

1/41004 100

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 28

Example

Write the expression below as a single radical. Assume that all variables represent nonnegative values.

Solution

4 x

4 x 1/2 1/4( )x(1/2)(1/4)x1/8x

8 x