Copyright © 2011 Pearson Education, Inc. Rational Exponents, Radicals, and Complex Numbers CHAPTER 10.1Radical Expressions and Functions 10.2Rational Exponents

Copyright © 2011 Pearson Education, Inc.

Rational Exponents, Radicals, and Complex Numbers

CHAPTER

10.1 Radical Expressions and Functions10.2 Rational Exponents10.3 Multiplying, Dividing, and Simplifying Radicals10.4 Adding, Subtracting, and Multiplying Radical

Expressions10.5 Rationalizing Numerators and Denominators of

Radical Expressions10.6 Radical Equations and Problem Solving 10.7 Complex Numbers

1100


Radical Expressions and Functions10.110.1

1. Find the nth root of a number.2. Approximate roots using a calculator.3. Simplify radical expressions.4. Evaluate radical functions.5. Find the domain of radical functions.6. Solve applications involving radical functions.

Slide 10- 3Copyright © 2011 Pearson Education, Inc.

Evaluating nth rootsWhen evaluating a radical expression , the sign of a

and the index n will determine possible outcomes.If a is nonnegative, then , where and bn = a.If a is negative and n is even, then there is no real-number root.If a is negative and n is odd, then , where b is negative and bn = a.

n a

n a b 0b

n a b

nth root: The number b is an nth root of a number a if bn = a.


Example 1

Evaluate each root, if possible.a.

b.

c.

169

Solution 169 13

0.49

Solution 0.49 0.7

100

Solution 100 is not a real number because there is no real number whose square is –100.


continued

Evaluate each root, if possible.d.

e.

f.

144

Solution 144 12

49

144

Solution 49

144

49

144

7

12

3 27

Solution 3 27 3


continued

Evaluate each root, if possible.g.

h.

3 27

Solution 3 27

4 81

Solution 4 81 3

3


Some roots, like are called irrational because we cannot express their exact value using rational numbers. In fact, writing with the radical sign is the only way we can express its exact value. However, we can approximate using rational numbers.

3

3

3

Approximating to two decimal places:

Approximating to three decimal places:

2 1.41

2 1.414

Note: Remember that the symbol, , means “approximately equal to.”


Example 2

Approximate the roots using a calculator or table in the endpapers. Round to three decimal places.a.

b.

c.

18

Solution

32

Solution 32 5.657

18 4.243

3 56

Solution 3 56 3.826


Example 3

Find the root. Assume variables represent nonnegative values.

b.

c.

d.

4y Solution

636m Solution

4y 2y

636m 36m

Because (y2)2 = y4.

Because (6m3)2 = 36m6.

10

4

36

25

x

ySolution

10 5

4 2

36 6

25 5

x x

y y


continued

Find the root. Assume variables represent nonnegative values.

e.

f.

93 y Solution

164 81x Solution

93 y

164 81x 43x

3y


Example 4

Find the root. Assume variables represent any real number.

a.

b.

c.

14y Solution

1036y Solution

14y 7y

56 y1036y

2( 3)n Solution 3n 2( 3)n


continued

Find the root. Assume variables represent any real number.

d.

e.

c.

1249y Solution

3 927n Solution

1249y 67 y

33n

33 ( 4)w Solution

3 927n

33 ( 4)w 4w


Radical function: A function containing a radical expression whose radicand has a variable.

Example 5a

Solution

Given f(x) = find f(3). 5 8,x

To find f(3), substitute 3 for x and simplify.

3 5 3 8f 15 8 7


Example 6Find the domain of each of the following. a.

b.

8f x x

Solution Since the index is even, the radicand must be nonnegative.

Solution The radicand must be nonnegative.

8 0x 8x

3 9f x x

3 9 0x 3 9x

3x

Domain: 8 , or [8, )x x

Domain: 3 , or ( ,3]x x

Conclusion The domain of a radical function with an even index must contain values that keep its radicand nonnegative.


Example 7

If you drop an object, the time (t) it takes in seconds to fall d feet is given by . Find the time ittakes for an object to fall 800 feet.

16dt

Understand We are to find the time it takes for an object to fall 800 feet.

Plan Use the formula , replacing d with 800. 16dt

Execute 80016t

50t

7.071t

Replace d with 800.

Divide within the radical.

Evaluate the square root.


continued

Answer It takes an object 7.071 seconds to fall 800 feet.

Check We can verify the calculations, which we will leave to the viewer.


For which square root is –12.37 the approximation for?

a)

b)

c)

d)

3.517

3.517

153

153


For which square root is –12.37 the approximation for?

a)

b)

c)

d)

3.517

3.517

153

153


Evaluate.

a) 0.2

b) 0.02

c) 0.002

d) 0.0002

0.0004


Evaluate.

a) 0.2

b) 0.02

c) 0.002

d) 0.0002

0.0004


Find the domain of f(x) = .

a)

b)

c)

d)

4 16x

4 , or ( , 4]x x

4 , or [4, )x x

4 , or [ 4, )x x

4 , or ( , 4]x x


Find the domain of f(x) = .

a)

b)

c)

d)

4 16x

4 , or ( , 4]x x

4 , or [4, )x x

4 , or [ 4, )x x

4 , or ( , 4]x x


Rational Exponents10.210.2

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.


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.


Example 1

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


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


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


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


Example 2

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)


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


Negative Rational Exponents

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

//

1,m n

m na

a


Example 3

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


continued

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

Solutionc.

1/2

1/2

25 1

36 2536

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


Example 4

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


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


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


Example 5a

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.


Example 5b


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.


Example 5c


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


Example 5d


Solution

Add the exponents.

2/5 3/53 5y y

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

1/515y


Example 5e


Solution

27/8m

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


Example 5f


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


Example 5g


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


Example 6

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


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


Example 7

Perform 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


continued

Perform the indicated operations. Write the result using a radical.

Solution

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

2/4 1/45 4

1/425 4 1/4(25 4)

1/41004 100


Example 8

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


Simplify.

a)

b)

c)

d)

3/ 41/ 2 2 /3x y

3/8 1/ 4x y

3/8 3/ 4x y

3/8 1/ 2x y

3/ 4 1/ 2x y


Simplify.

a)

b)

c)

d)

3/ 41/ 2 2 /3x y

3/8 1/ 4x y

3/8 3/ 4x y

3/8 1/ 2x y

3/ 4 1/ 2x y


Simplify.

a) 5

b) 25

c) 25

d) 5

3 325 5


Simplify.

a) 5

b) 25

c) 25

d) 5

3 325 5


Simplify.

a) 4

b)

c) 4

d)

2/38

1

4

1

4


Simplify.

a) 4

b)

c) 4

d)

2/38

1

4

1

4


Multiplying, Dividing, and Simplifying Radicals10.310.3

1. Multiply radical expressions.2. Divide radical expressions.3. Use the product rule to simplify radical expressions.


Product Rule for Radicals

If both and are real numbers, then

.n n na b a b

n a n b


Example 1

Find the product and simplify. Assume all variables represent positive values. a. b.

Solutiona. b.

4 9 7 y

4 9 4 9

36

6

7 y 7y


continued

Find the product and simplify. Assume all variables represent positive values. c. d.

Solution

c. d.

4 42 8 3 34 5x x

4 42 8 4 2 8

4 16

2

3 34 5x x 3 4 5x x

3 220x


continued

Find the product and simplify. Assume all variables represent positive values. e. f.

Solution

e. f.

6

5

y

w

6

5

y

w

6

5

y

w

6

5

y

w

5 97 7y y

5 97 7y y 5 97 y y

147 y

2y


continued

Find the product and simplify. Assume all variables represent positive values. g.

Solution

g.

2 2x x

2 2x x 2 2x x

4x

2x


Raising an nth Root to the nth PowerFor any nonnegative real number a,

, where 0.n

nn

a ab

bb

Quotient Rule for RadicalsIf both and are real numbers, then

.n

n a a

n a n b


Example 2Simplify. Assume variables represent positive values.a.

Solution

147

3

147

349 7

147

3

b.

b.

11

49

a. 11

49

11

49 11

7

c. 36

15

x

c. 36

15

x

3

3 6

15

x

3

2

15

x


continuedSimplify. Assume variables represent positive values.d.

Solution

5

1024

x

5

1024

x 5

4

x

5

5 1024

x

e.

e.

5

5

12

4

d.5

5

12

45

12

4 5 3


Simplifying nth Roots To simplify an nth root, 1. Write the radicand as a product of the greatest

possible perfect nth power and a number or an expression that has no perfect nth power factors.

2. Use the product rule when a is the perfect nth power.

3. Find the nth root of the perfect nth power radicand.

n n nab a b


Example 3

Simplify. a. b.

Solution

80

80 16 5

16 5

4 5

6 98

6 49 2

6 49 2

6 7 2

Solution

6 98

42 2


continued

Simplify. c. d.

Solution

34 448

3 34 64 7

34 4 7

316 7

Solution34 448

45 48

45 48 45 16 3

4 45 16 3

45 2 3

410 3


Example 4a

Simplify the radical using prime factorization.

Solution

7 7 7 2

7 7 2

7 14

686Write 686 as a product of its prime factors.

The square root of the pair of 7s is 7.

Multiply the prime factors in the radicand.


continued

Simplify the radical using prime factorization. b. c.

b.

Solution

3 2 2 5 5 5

35 2 2

35 4

3 500

3 500 4 810

c. 4 810 4 2 3 3 3 3 5

43 2 5

43 10


Example 5a

Simplify.

Solution

532x

416 2 x x

416 2x x

24 2 x x

532x The greatest perfect square factor of 32x5 is 16x4.

Use the product rule of square roots to separate the factors into two radicals.

Find the square root of 16x4 and leave 2x in the radical.


Example 5b

Simplify

Solution

42 96 .a b

42 16 6 a b

42 16 6a b

22 4 6 a b

The greatest perfect square factor of 96a4b is 16a4.

Use the product rule of square roots to separate the factors into two radicals.

Find the square root of 16a4 and leave 6b in the radical.

42 96a b

28 6 a b Multiply 2 and 4.


continued

Simplify. c. d.

Solution

3 103y y

3 103y y 3 93 y y y

3 93 3 y y y

3 3 3y y y

11 145 486x y

10 10 45 243 2 x x y y

Solution

11 145 486x y

6 3y y

10 10 45 5243 2x y x y

2 2 453 2x y xy


Example 6

Find the product or quotient and simplify the results. Assume that variables represent positive values.a. b.

Solution

5 8 5 44 5 5 50x x

Solution5 8 40

4 10

2 10

5 44 5 5 50x x5 44 5 5 50x x

920 250x820 25 10 x x

420 5 10x x 4100 10x x


continuedFind the product or quotient and simplify the results. Assume that variables represent positive values.c. d.

Solution

300

4

9 6

5

9 245

3 5

a b

a bSolution

300

4

300

4

75

25 3

9 6

5

9 245

3 5

a b

a b

9 6

5

2453

5

a b

a b

4 53 49a b4 43 49a b b

2 23 7a b b 2 221a b b5 3


Simplify. Assume all variables represent nonnegative numbers.

a)

b)

c)

d)

2 33 9m m

3 3m m

23 3m m

627m

2 33 3m m


Simplify. Assume all variables represent nonnegative numbers.

a)

b)

c)

d)

2 33 9m m

3 3m m

23 3m m

627m

2 33 3m m


Simplify.

a)

b)

c)

d)

486

6 9

3 54

9 6

18 27


Simplify.

a)

b)

c)

d)

486

6 9

3 54

9 6

18 27


Adding, Subtracting, and Multiplying Radical Expressions10.410.4

1. Add or subtract like radicals.2. Use the distributive property in expressions containing

radicals.3. Simplify radical expressions that contain mixed

operations.


Like radicals: Radical expressions with identical radicands and identical root indices.

Adding Like Radicals

To add or subtract like radicals, add or subtract the coefficients and leave the radical parts the same.


Example 1

Add or subtract. a. b.

6 7 3 7

Solution

3 34 5 2 5

a.

b.

6 7 3 7 (6 3) 7

3 34 5 2 5 3(4 2) 5

36 5

9 7


continued

Simplify.c. d.

3 37 5 12 5x x

Solution

14 5 11 2 11 5 5

c.

d.

3 37 5 12 5x x 35 5x Combine the like radicals by subtracting the coefficients and keeping the radical.

14 5 11 2 11 5 5

(14 5) 5 (2 1) 11

9 5 11

Regroup the terms.


Example 2

Add or subtract. a. b.

28 7

Solution

3 34 135 2 5

a.

b.

4 7 7 Factor 28.28 7

2 7 7 3 7 Combine like radicals.

Simplify.

3 34 135 2 5 3 34 27 5 2 5 3 34 3 5 2 5

3 312 5 2 5 310 5


continued

c.

Solution

5 5 563 112 28x x x

c.5 5 563 112 28x x x 4 4 49 7 16 7 4 7x x x x x x

2 2 23 7 4 7 2 7x x x x x x 23 7x x


Example 3a

Find the product.

Solution

3 6 5 7 7

3 6 5 7 7 3 6 5 3 6 7 7

3 30 21 42

Use the distributive property.

Multiply.


Example 3c

Find the product.

Solution

Use the product rule.

4 5 2 5 5 2 .

4 5 2 5 5 2

4 5 5 5 4 5 2 5 2 5 2 2

4 5 20 10 10 5 2

20 20 10 10 10

10 19 10

Use the distributive property.

Find the products.

Combine like radicals.


Example 3d

Find the product.

Solution

4 3 7x y x y

4 3 7x y x y

4 3 4 7 3 7x x x y y x y y

12 28 3 7x xy xy y

12 25 7x xy y


Example 3e

Find the product.

Solution

2

5 3

2 2

5 2 5 3 3

5 2 15 3

8 2 15

Simplify.

2

5 3 Use (a – b)2 = a2 – 2ab – b2.


Example 4a

Find the product.

Solution

8 3 8 3

228 3

61

8 3 8 3

64 3 Simplify.

Use (a + b)(a – b) = a2 – b2.


Example 4b

Find the product.

Solution

7 2 3 7 2 3

2 2

7 2 3

7 12

7 2 3 7 2 3

7 4 3

5


Example 5

Simplify.a.

Solution

9650

3

a. b.

2 14 3 21

9650

3 32 25 2

16 2 5 2

16 2 5 2

4 2 5 2

9 2

2 14 3 21

2 14 3 21 28 63

4 7 9 7 2 7 3 7

5 7

b.


Simplify.

a)

b)

c)

d)

2 12 5 27 48 2 3

9 3

17 3

71 3

25 3


Simplify.

a)

b)

c)

d)

2 12 5 27 48 2 3

9 3

17 3

71 3

25 3


Multiply.

a)

b)

c)

d)

3 2 3 2 5 3

9 15 6 3 2

6 3 2 15 3 15 6

5 14 6 15 3

25 3


Multiply.

a)

b)

c)

d)

3 2 3 2 5 3

9 15 6 3 2

5 14 6 15 3

25 3

6 3 2 15 3 15 6


Rationalizing Numerators and Denominators of Radical Expressions10.510.5

1. Rationalize denominators.2. Rationalize denominators that have a sum or difference

with a square root term.3. Rationalize numerators.


Example 1a

Rationalize the denominator.

Solution

Simplify.

8

5

Multiply by

8 5

25

8

5

8 5

5 5 5

.5

8 5

5


Example 1b


SolutionUse the quotient rule for square roots to separate the numerator and denominator into two radicals.

2

32

3

2

3

2 3

3 3

6

9

6

3

Multiply by 3

.3

Simplify.

Warning: Never divide out factors common to a radicand and a number not under a radical.


Example 1c


Solution

5

3x

3

5

3

3

xx

x

5

3x

2

5 3

9

x

x

5 3

3

x

x


Rationalizing DenominatorsTo rationalize a denominator containing a single nth root, multiply the fraction by a well chosen 1 so that the product’s denominator has a radicand that is a perfect nth power.


Example 2a

Rationalize the denominator. Assume that variables represent positive values.

Solution

3

5

3

3

5

3 33

35

3

9

9

3

3

5 9

27

35 9

3


Example 2b


Solution

3

3

w

z

3

3

w

z

2

3

3

23

3 zw

z z

3 2

3 3

wz

z

3 2wz

z


Example 2c


Solution

32

7

16x

32

7

16x

3

3 2

7

16x

3

3 2

3

3

7

1 46

4

x

x

x

3

3 3

28

64

x

x

3 28

4

x

x


Rationalizing a Denominator Containing a Sum or DifferenceTo rationalize a denominator containing a sum or difference with at least one square root term, multiply the fraction by a 1 whose numerator and denominator are the conjugate of the denominator.


Example 3a

Rationalize the denominator and simplify. Assume variables represent positive values.

Solution

7

3 57

3 53 5

3 55 3

7

2 2

7( 3 5)

( 3) (5)

7 3 35

3 25

7 3 35

22

1(7 3 35)

1(22)

35 7 3

22


Example 3b


Solution

12 5

11 312 5

11 3

11 3

11 3

12 5

11 3

2 2

12 5 11 3

( 11) ( 3)

12 55 12 15

11 3

12 55 12 15

8

4(3 55 3 15)

8

3 55 3 15

2


Example 3c


Solution

4

3x

4

3x 4 3

3 3

x

x x

2 2

4 3

3

x

x

4 12

9

x

x


Example 4a

Rationalize the numerator. Assume variables represent positive values.

Solution

3

8

x

3

8

x

3

3 3

8 x

x x

29

8 3

x

x

3

8 3

x

x


Example 4b

Rationalize the numerator. Assume variables represent positive values.

Solution

5 3

6

x

5 3

6

x 5 3 5 3

6 5 3

x x

x

225 3

6 5 3

x

x

25 3

30 6 3

x

x



a)

b)

c)

d)

8

3

2 3

33 2

3

2 6

3

3 6

3



a)

b)

c)

d)

8

3

2 3

33 2

3

2 6

3

3 6

3



a)

b)

c)

d)

5

13 7

5 13 5 7

6

5 13 5 7

6

13 7

5

13 7

5



a)

b)

c)

d)

5

13 7

5 13 5 7

6

5 13 5 7

6

13 7

5

13 7

5


Radical Equations and Problem Solving10.610.6

1. Use the power rule to solve radical equations.


Power Rule for Solving EquationsIf both sides of an equation are raised to the same integer power, the resulting equation contains all solutions of the original equation and perhaps some solutions that do not solve the original equation. That is, the solutions of the equation a = b are contained among the solutions of an = bn, where n is an integer.

Radical equation: An equation containing at least one radical expression whose radicand has a variable.


Example 1

Solve. a. b.

Solution a.

12y 3 4x

2212y

144y

144 12

12 12

Check

True

b. 3 33 4x

64x

Check3 64 4

4 4 True


Solution

Example 2a Solve.5 6x

5 6x

225 (6)x

5 36x

41x

Check:

The number 41 checks. The solution is 41.

5 6x

41 5 6

36 6

6 6


Solution

Example 2b Solve.3 4 2x

3 4 2x

333 4 ( 2)x

4 8x

4x

Check:

True. The solution is 4.

3 4 2x

3 4 4 2

3 8 2

2 2


Solution

Example 2c Solve.4 1 5x

4 1 5x

224 1 ( 5)x

4 1 25x

4 24x

Check:

False, so 6 is extraneous. This equation has no real number solution.

4 1 5x

4(6) 1 5

25 5

5 5

6x


Solution

Example 3a Solve.4 60x x

4 60x x

2 2

4 60x x

224 60x x

16 60x x

15 60x

4x

Check:

The number 4 checks. The solution is 4.

4 60x x

4 4 4 60

4 2 64

8 8


Example 4Solve.Solution

5 7x x

5 7x x

225 7x x

2 10 25 7x x x 2 11 25 7x x 2 11 18 0x x

( 2)( 9) 0x x 2 0 or 9 0x x

2x 9x

Square both sides.

Use FOIL.

Subtract x from both sides.

Factor.

Use the zero-factor theorem.

Subtract 7 from both sides.


continued

Checks 2x 9x

5 7x x

2 5 2 7

3 9

3 3 False.

9 5 9 7

4 16

4 4 True.

Because 2 does not check, it is an extraneous solution. The only solution is 9.


Example 5a

Solve.

SolutionCheck

This solution does not check, so it is an extraneous solution. The equation has no real number solution.

4 6x

4 6x

2x

2 2

2x

4x

4 6x

4 4 6

2 4 6 2 6


Example 5b

Solve 4 3 3 5.x

Solution

Check

The solution set is 13.

4 3 3 5x 4 3 2x

444 3 2x

3 16x

13x

4 3 3 5x

4 13 3 3 5

4 16 3 5

2 3 5

5 5


Example 6

Solve 16 8x x

SolutionCheck

There is no solution.

16 8x x

2 2

16 8x x

16 8 8 64x x x x

16 16 64x x x

48 16 x

16 8x x

9 16 9 8

25 3 8

5 11

3 x 2 2( 3) ( )x 9 x


Solving Radical EquationsTo solve a radical equation,1. Isolate the radical if necessary. (If there is more

than one radical term, isolate one of the radical terms.)

2. Raise both sides of the equation to the same power as the root index of the isolated radical.

3. If all radicals have been eliminated, solve. If a radical term remains, isolate that radical term and raise both sides to the same power as its root index.

4. Check each solution. Any apparent solution that does not check is an extraneous solution.


Solve.

d) no solution

a) 6

b) 8

c) 9

5 4 2x x


Solve.

d) no solution

a) 6

b) 8

c) 9

5 4 2x x


Solve.

d) no real-number solution

a) 2

b) 4

c)

4 10 2a a

12

5


Solve.


a) 2

b) 4

c)

4 10 2a a

12

5


Solve.


a) 3, 4

b) 3

c) 4

2 5 16x x


Solve.


a) 3, 4

b) 3

c) 4

2 5 16x x


Complex Numbers10.710.7

1. Write imaginary numbers using i.2. Perform arithmetic operations with complex numbers.3. Raise i to powers.


Imaginary unit: The number represented by i, where and i2 = 1.

Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.

1i


Example 1

Write each imaginary number as a product of a real number and i.a. b. c.

Solutiona. b. c.

16 21 32

16 21 32

1 16

1 16 4i

4i

1 21

1 21 21i

1 32

1 32

16 2i 4 2i


Rewriting Imaginary NumbersTo write an imaginary number in terms of the imaginary unit i,1. Separate the radical into two factors, 2. Replace with i.3. Simplify

n

1 .n 1.n


Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.


Example 2a

Add or subtract. (9 + 6i) + (6 – 13i)

SolutionWe add complex numbers just like we add polynomials—by combining like terms.

(9 + 6i) + (6 – 13i) = (9 + 6) + (6i – 13i ) = –3 – 7i


Example 2b

Add or subtract. (3 + 4i) – (4 – 12i)

SolutionWe subtract complex numbers just like we subtract polynomials.

(3 + 4i) – (4 – 12i) = (3 + 4i) + (4 + 12i) = 7 + 16i


Example 3Multiply.a. (8i)(4i) b. (6i)(3 – 2i)

Solution a. (8i)(4i) b. (6i)(3 – 2i) 232i

)132(

32

218 12i i

18 1 ( 12 )i

18 12i

12 18i


continuedMultiply.c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i)

Solution c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i)

227 9 12 4i i i

27 3 4 )1(i

27 3 4i

31 3i

249 14 14 4i ii

49 4 1( )

49 4

53


Complex conjugate: The complex conjugate of a complex number a + bi is a – bi.


Example 4a

Divide. Write in standard form.

Solution Rationalize the denominator.

7

3i

7

3i7

3

i

i i

2

7

3

i

i

7

3( 1)

i

7

3

i

7

3

i


Example 4b

Divide. Write in standard form.

Solution Rationalize the denominator.

3 5

5

i

i

3 5

5

i

i

53

5

5

5

i

i

i

i

2

2

15 3 25 5

25

i i i

i

15 3 25 5( 1)

25 ( 1)

i i

15 3 25 5

25 1

i i

10 28

26

i

10 28

26 26

i

5 14

13 13

i


Example 5

Simplify.

Solution

40 33a. b. i i

1040 4 10a. = = 1 i i = 1 Write i40 as (i4)10.

33 32b. = i i i

84 = i i

= 1 i

= i

Write i32 as (i4)8.

Replace i4 with 1.


Simplify. (4 + 7i) – (2 + i)

a) 2 + 7i2

b) 2 + 8i

c) 6 + 6i

d) 6 + 8i


Simplify. (4 + 7i) – (2 + i)

a) 2 + 7i2

b) 2 + 8i

c) 6 + 6i

d) 6 + 8i


Multiply. (4 + 7i)(2 + i)

a) 15 + 10i

b) 1 + 10i

c) 15 + 18i

d) 15 + 18i


Multiply. (4 + 7i)(2 + i)

a) 15 + 10i

b) 1 + 10i

c) 15 + 18i

d) 15 + 18i


Write in standard form.

a)

b)

c)

d)

4

2 3

i

i

5 14

13 13

i

5 14

13 13

i

11 14

13 13

i

11 14

13 13

i


Write in standard form.

a)

b)

c)

d)

4

2 3

i

i

5 14

13 13

i

5 14

13 13

i

11 14

13 13

i

11 14

13 13

i

Documents

Copyright © 2011 Pearson Education, Inc. Rational Exponents, Radicals, and Complex Numbers CHAPTER 10.1Radical Expressions and Functions 10.2Rational Exponents