CHAPTER Solutions Key 11 Exponential and Radical … · Solutions Key 11 Exponential and Radical...

Solutions KeyExponential and Radical Functions11

CHAPTER

xzARE YOU READY, PAGE 763

1. B; like terms: terms that contain the same variable raised to the same power

2. F; square root: one of two equal factors of a number

3. C; domain: the set of first elemtns of a relation

4. E; perfect square: a number whose positive square root is a whole number

5. D; exponent: a number that tells how many times a base is used as a factor

6. 16 7. 1 8. 63 9. 375

10. 243 11. -28 12. 320 13. 147

14. y = 8 15. y = x + 3

16. y = x 2 - 4 17. y = x 2 + 2

18. 0.5 19. 0.25 20. 0.152 21. 2.0

22. 0.019 23. 0.003 24. 0.001 25. 0.0104

26. 6; 6 · 6 = 36 27. 9; 9 · 9 = 81

28. 5; 5 · 5 = 25 29. 8; 8 · 8 = 64

30. h 2 = 3 2 + 4 2 h 2 = 25 h = 5 cm

31. h 2 = 12 2 + 5 2 h 2 = 169 h = 13 in.

32. h 2 = 6 2 + 8 2 h 2 = 100 h = 10 ft

33. 5(2m - 3) = 5 · 2m - 5 · 3 = 10m - 15

34. 3x(8x + 9) = 3x · 8x + 3x · 9 = 24 x 2 + 27x

35. 2t(3t - 1) = 2t · 3t - 2t · 1 = 6 t 2 - 2

36. 4r(4r - 5) = 4r · 4r - 4r · 5 = 16 r 2 - 20r

11-1 GEOMETRIC SEQUENCES, PAGES 766–771

CHECK IT OUT! PAGE 767

1a. 80, -160, 320; (-10) ÷ 5 = -2, 20 ÷ (-10) = -2, (-40) ÷ 20 = -2So, the common ratio is -2.(-40) · (-2) = 80, 80 · (-2) = -160,and (-160) · (-2) = 320

b. 216, 162, 121.5; 384 ÷ 512 = 3 __ 4

, 288 ÷ 384 = 3 __ 4

,

So, the common ratio is 3 __ 4 .

288 · 3 __ 4

= 216, 216 · 3 __ 4

= 162,

and 162 · 3 __ 4

= 121.5

2. a n = a 1 r n - 1

a 8 = 1000 ( 1 __ 2 )

7

a 8 = 7.8125

3. a n = a 1 r n - 1

a 10 = 10,000 ( 4 __ 5 )

9

a 10 = 1342.18;

$1342.18

THINK AND DISCUSS, PAGE 768

1. Possible answer: Divide each term after the first by the preceding term. If the quotients are all the same, the sequence is geometric.

2. Possible answer:

EXERCISES, PAGES 769–771GUIDED PRACTICE, PAGE 769

1. common ratio: the value that each term is multiplied by to get the next term.

Copyright © by Holt, Rinehart and Winston. 435 Holt Algebra 1All rights reserved.

2. 32, 64, 128; 4 ÷ 2 = 2, 8 ÷ 4 = 2, 16 ÷ 8 = 2So, the common ratio is 2. Then, 16 · 2 = 32, 32 · 2 = 64,and 64 · 2 = 128

3. 25, 12.5, 6.25; 200 ÷ 400 = 1 __ 2,

100 ÷ 200 = 1 __ 2 , 50 ÷ 100 = 1 __

2


Then, 50 · 1 __ 2 = 25, 25 · 1 __

2 = 12.5,

and 12,5 · 1 __ 2 = 6.25

4. 324, -972, 2916; (-12) ÷ 4 = -3, 36 ÷ (-12) = -3, (-108) ÷ 36 = -3So, the common ratio is -3. Then, (-108) · (-3) = 324, 324 · (-3) = -972,and (-972) · (-3) = 2916

5. a n = a 1 r n - 1

a 10 = 1 · 10 10 - 1

a 10 = 1,000,000,000

6. a n = a 1 r n - 1

a 11 = 3 · 2 11 - 1

a 11 = 3072

7. 32 ___ 64

= 1 __ 2 ; 16 ___

32 = 1 __

2

a n = a 1 r n - 1

a 5 = 64 · ( 1 __ 2 )

4

a 5 = 4

PRACTICE AND PROBLEM SOLVING, PAGES 769–770

8. -1250, 6250, -31,250; 10 ___ -2 = -5;

-50 ____ 10

= -5; 250 _____ -50 = -5

So, the common ratio is -5.Then, 250 · (-5) = -1250, -1250 · (-5) = 6250,and 6250, · (-5) = -31 250

9. 162, 243, 364.5; 48 ___ 32

= 3 __ 2 ; 72 ___

48 = 3 __

2 ; 108 ____

72 = 3 __

2


Then, 108 ( 3 __ 2 ) = 162, 162 ( 3 __

2 ) = 243,

and 243 ( 3 __ 2 ) = 364.5

10. 256, 204.8, 163.84; 500 ____ 625

= 4 __ 5 ; 400 ____

500 = 4 __

5 ; 320 ____

400 = 4 __

5


Then, 320 ( 4 __ 5 ) = 256, 256 ( 4 __

5 ) = 204.8,

and 204.8 ( 4 __ 5 ) = 163.84

11. 2058, 14 406, 100 842; 42 ___ 6 = 7; 294 ____

42 = 7

So, the common ratio is 7.Then, 294 · 7 = 2058, 2058 · 7 = 14, 406and 14, 406 · 7 = 100, 842

12. 96, -192, 384; - 12 ___ 6 = -2; 24 ____ -12

= -2; -48 ____ 24

= -2

So, the common ratio is -2.Then, -48 (-2) = 96, 96 (-2) = -192, and-192 (-2) = 384

13. 5 ___ 32

, 5 ____ 128

, 5 ____ 512

; 10 ___ 40

= 1 __ 4 ; 5 __

2 ÷ 10 = 1 __

4 ; 5 __

8 ÷ 5 __

2 = 1 __

4


Then, ( 5 __ 8 ) ( 1 __

4 ) = 5 ___

32 , ( 5 ___

32 ) ( 1 __

4 ) = 5 ____

128 , and

( 5 ____ 128

) ( 1 __ 4 ) = 5 ____

512

14. a n = a 1 r n - 1 a 5 = 18 · (3.5) 5 - 1 a 5 = 2701.125

15. 100 _____ 1000

= 1 ___ 10

; 10 ____ 100

= 1 ___ 10

; 1 ___ 10

= 1 ___ 10

a n

= a 1 r n - 1

a 14 = 1000 · 0.1 14 - 1 a 14 = 0.0000000001 or a 14 = 1 × 10 -10

16. 83.9 m; 320 ____ 400

= 4 __ 5 ; 256 ____

320 = 4 __

5

a n = a 1 r n - 1

a 8 = 400 ( 4 __ 5 )

8 - 1

a 8 = 83.9

17. 20, 40, 80, 160; 40 ___ 20

= 2, so the common ratio is 2;

40 · 2 = 80 and 80 · 2 = 160

18. 2, 6, 18, 54; 18 ___ 6 = 3, so the common ratio is 3;

6 __ 3 = 2 and 18 · 3 = 54

19. 9, 3, 1, 1 __ 3 ; 3 __

9 = 1 __

3 ; 1 __

3 = 1 __

3

So the common ratio is 1 __ 3 ; 1 · 1 __

3 = 1 __

3

20. 3, 12, 48, 192, 768; 12 ___ 3 = 4, so the common ratio

is 4; 12 · 4 = 48 and 192 · 4 = 768

21. 7, 1, 1 __ 7 , 1 ___

49 , 1 ____

343 ; The common ratio is 1 __

7 ;

1 · 1 __ 7 = 1 __

7 and 1 __

7 · 1 __

7 = 1 ___

49

22. 400, 100, 25, 25 ___ 4 ; 25 ____

100 = 1 __

4 , so the common ratio

is 1 __ 4 .

Then, 100 ÷ 1 __ 4 = 25 and 25 · 1 __

4 = 25 ___

4

23. -3, 6, -12, 24, -48; 24 ____ -12 = -2, so the common

ratio is -2.Then, -3 · (-2) = 6 and 24 · (-2) = -48

24. 1 __ 9 , - 1 __

3 , 1, -3, 9; - 3 __

1 = -3; 9 ___ -3

= -3

So the common ratio is -3.

Then, 1 ÷ -3 = - 1 __ 3 and - 1 __

3 ÷ -3 = 1 __

9

25. 1, 17, 289, 4913; 17 ___ 1 = 17; 289 ____

17 = 17

So the common ratio is 17.Then, 289 · 17 = 4913


26. 10 ___ 2 = 5; 50 ___

10 = 5; 250 ____

50 = 5

The common ratio is 5; yes.

27. 15 ___ 5 = 1 __

3 ; 5 __

3 ÷ 5 = 1 __

3 ; 5 __

9 ÷ 5 __

3 = 1 __

3

The common ratio is 1 __ 3 ; yes.

28. 18 ___ 6 = 3; 24 ___

18 = 4 __

3 ; 38 ___

24 = 19 ___

12

There is no common ratio; no.

29. 3 __ 9 = 1 __

3 ; -1 ___

3 = - 1 __

3 ; -5 ___ -1

= 5


30. 21 ___ 7 = 3; 63 ___

21 = 3; 189 ____

63 = 3

The common ratio is 3; yes.

31. 1 __ 4 = 1 __

4 ; -2 ___

1 = -2; -4 ___ -2

= 2


32a. 2 __ 1 = 2; 4 __

2 = 2; 8 __

4 = 2

Plan 2 is a geometric sequence with common ratio 2.

b. Possible answer: Plan 1; Under Plan 2, the cost for the 10th week alone is $512, which is more than the cost for the entire summer under Plan 1.

33a. a n = a 1 r n - 1 a 7 = 0.02 · 2 6 a 7 = 1.28 cm

b. a n = a 1 r n - 1 a 12 = 0.02 · 2 11 a 12 = 40.96 cm

34. a 1 = 3

a 2 = 3 (2) 1 = 6

a 3 = 3 (2) 2 = 12

a 4 = 3 (2) 3 = 24

35. a 1 = -2

a 2 = -2 (4) 1 = -8

a 3 = -2 (4) 2 = -32

a 4 = -2 (4) 3 = -128

36. a 1 = 5

a 2 = 5 (-2) 1 = -10

a 3 = 5 (-2) 2 = 20

a 4 = 5 (-2) 3 = -40

37. a 1 = 2

a 2 = 2 (2) 1 = 4

a 3 = 2 (2) 2 = 8

a 4 = 2 (2) 3 = 16

38. a 1 = 2

a 2 = 2 (5) 1 = 10

a 3 = 2 (5) 2 = 50

a 4 = 2 (5) 3 = 250

39. a 1 = 12

a 2 = 12 ( 1 __ 4 )

1 = 3

a 3 = 12 ( 1 __ 4 )

2 = 3 __

4

a 4 = 12 ( 1 __ 4 )

3 = 3 ___

16

40. Each term is multiplied by 2 n - 1 , where n is the term number. For example, begin with the geometric sequence 4, 12, 36, 108. ..., where r = 3. If r is doubled to 6, the sequence becomes 4, 24, 144, 864, ....

41a. Stage 0 Stage 1:

Stage 2: Stage 3:

b. Stage Squares

0 1

1 4

2 16

3 64

c. 4 __ 1 = 4; 16 ___

4 = 4; 64 ___

16 = 4

yes; r = 4

d. r = 4 and a 1 = 4 a n = a 1 r n - 1 a n = 4 (4) n - 1 a n = 4 n

42. Divide each term by the preceeding term to find the value of r. Then use the formula a n = a 1 r n - 1 , where a 1 is the first term of the sequence.

43a. 3300 _____ 3000

= 1.1; 3630 _____ 3300

= 1.1

a 4 = 3630 · 1.1 = $3993 a 5 = 3993 · 1.1 = $4392.30

b. 3300 _____ 3000

= 1.1; 3630 _____ 3300

= 1.1

The common ratio is 1.1.

c. $2727.27; divide tuition 3 years ago ($3000) by 1.1, the common ratio.


TEST PREP, PAGE 771

44. D: 10 ___ 5 = 2; 20 ___

10 = 2; 40 ___

20 =2; there is a common

ratio.

45. J; since r = -4 and a 1 = 2,

( -8 ___ 2 = -4; 32 ___ -8

= -4; -128 _____ 32

= -4)

a n = 2 (-4) n - 1

46. C; r = 2 and A 1 = 55

A n = A a r n - 1

A 7 = A 1 r 6 A 7 = 3520 Hz

CHALLENGE AND EXTEND, PAGE 771

47. x 2 __ x = x; x 3 __ x 2

= x

r = x and a 1 = x;

a 4 = x (x) 3 = x 4

a 5 = x (x) 4 = x 5

a 6 = x (x) 5 = x 6

48. 6 x 3 ___ 2 x 2

= 3x; 18 x 4 ____ 6 x 3

= 3x

r = 3x and a 1 = 2 x 2 ;

a 4 = 2 x 2 (3x) 3 = 54 x 5

a 5 = 2 x 2 (3x) 4 = 162 x 6

a 6 = 2 x 2 (3x) 5 = 486x 7

49. 1 __ y 2

÷ 1 __ y 3

= y; 1 __ y ÷ 1 __ y 2

= y

r = y and a 1 = 1 __ y 3

a 4 = 1 __ y 3

(y) 3 = 1

a 5 = 1 __ y 3

(y) 4 = y

a 6 = 1 __ y 3

(y) 5 = y 2

50. 1 ____ x + 1 ÷ 1 ______

(x+1) 2 = x + 1; 1 ÷ 1 ____

x+1 = x + 1

r = x + 1 and a 1 = 1 ______ (x+1) 2

a 4 = 1 _______ (x+1) 2

(x + 1) 3 = x + 1

a 5 = 1 ______ (x+1) 2

(x + 1) 4 = (x + 1) 2

a 6 = 1 ______ (x+1) 2

(x + 1) 5 = (x + 1) 3

51. a 10 = a 1 r 9

a 1 = a 10

___ r 9

a 1 = 0.78125 _______ (-0.5) 9

a 1 = -400

52. No; each term of the sequence is found by multiplying the previous term by the common

ratio 1 __ 2 . 1 __

2 of any positive number is always another

positive (nonzero) number.

53. a n = a 1 r n - 1

r n - 1 = a n

__ a 1

(0.4) n - 1 = 0.057344 ________ 14

(0.4) n - 1 = (0.4) 6 Then, n - 1 = 6 n = 7

54. Susanna assumed the sequence was geometric with r = 2. She used the formula to find a 8 = 128. Paul did not assume the sequence was geometric. Instead, he noticed a pattern of “add 1, add 2, and so on.” He continued this pattern by adding 3, adding 4, etc., until he got the 8th term of 29. Both could be considered correct because it was not specified what type of sequence was given.

SPIRAL REVIEW, PAGE 771

55. b - 4 > 6b - 4 + 4 > 6 + 4 b > 10

56. -12 + x ≤ -8-12 + 12 + x ≤ -8 + 12 x ≤ 4

57. c + 2 __ 3 < 1 __

3

c + 2 __ 3 - 2 __

3 < 1 __

3 - 2 __

3

c < - 1 __ 3

58. y < 2x - 4

59. 3x + y > 6 y > -3x + 6

60. -y ≤ 2x + 1 y ≥ -2x - 1


61. Vertical translation of +7;f(x) = x 2 - 3 + 7f(x) = x 2 + 4

62. Vertical translation of -2;f(x) = 2 x 2 + 6 - 2f(x) = 2 x 2 + 4Narrowing the graph.f(x) = a x 2 + 4, where a > 2.Possible answer: f(x) = 3 x 2 + 4

11-2 EXPONENTIAL FUNCTIONS, PAGES 772–778

CHECK IT OUT! PAGES 772−775

1. f(x) = 8 (0.75) x f(3) = 8 (0.75) 3 f(3) = 8(0.421875)f(3) = 3.375 in.

2a. No; as the x-values change by a constant amount, the y-values are not multiplied by a constant amount.

b. Yes; as the x-values change by a constant amount, the y-values are multiplied by a constant amount.

3a. y = 2 x b. y = 0.2 (5) x

4a. y = - 6 x b. y = -3 (3) x

5a. y = 4 ( 1 __ 4 )

x b. y = -2 (0.1) x

6. f(x) = 12,330 (0.869) x 2000 = 12,330 (0.869) x

x = log 0.869 ( 2 000 ______ 12,330

)

x ≈ 13; after about 13 yrs


1. Possible answer: Make a table of values. Use x-values that change by the same amount each time as you move down the column. Then divide each y-value, starting with the second row, by the y-value before it. The quotient is the common ratio.

2.

EXERCISES, PAGES 776−778GUIDED PRACTICE, PAGE 776

1. No; there is no variable in the exponent.

2. f(x) = 50,000 (0.975) x f(200) = 50,000 (0.975) 200 f(200) = 316; 316 lumens/ m 2

3. No; as the x-values increase by a constant value, the y-values are not multiplied by a constant value.

4. Yes; as the x-values increase by a constant value, the y-values are multiplied by a constant value.

5. y = 3 x 6. y = 5 x

7. y = 10 (3) x 8. y = 5 (2) x

9. y = -2 (3) x 10. y = -4 (2) x


11. y = -3 (2) x 12. y = 2 (3) x

6

3

x

y

05

13. y = - ( 1 __ 4 )

x 14. y = ( 1 __

3 )

x

15. y = 2 ( 1 __ 4 )

x 16. y = -2 (0.25) x

17. f(x) = 57.8 (1.02) x 200,000,000 = 57.8 (1.02) x x ≈ 63; about 2023 (63 years after 1960)

PRACTICE AND PROBLEM SOLVING, PAGES 776−778

18. f(x) = 27 ( 2 __ 3 )

x

f(4) = 27 ( 2 __ 3 )

4

f(4) = 27 ( 16 ___ 81

)

f(4) = 5 1 __ 3 ; 5 1 __

3 ft

19. y = 334 (0.976) x for x = 6,y = 334 (0.976) 6 y ≈ 289; 289 ft

20. y = 1.3 (1.41) x for x = 15,y = 1.3 (1.41) 15 y ≈ 225.02; 225.02 in./min

21. Yes; as the x-values change by a constant amount, the y-values are multiplied by a constant amount.

22. No; as the x-values change by a constant amount, the y-values are not multiplied by a constant amount.



25. y = 1.5 x 26. y = 1 _ 3 (3) x

27. y = 100 (0.7) x 28. y = -2 (4) x

29. y = -1 (5) x 30. y = - 1 _ 2 (4) x

31. y = 4 ( 1 _ 2 )

x 32. y = -2 ( 1 _

3 )

x

33. y = 0.5 (0.25) x

34. f(x) = 42 (1.41) x 1,000 = 42 (1.41) x x ≈ 9; about 2009

35. y = (3.1x + 7) 2 is not exponential since there is no variable in the exponent.

For y = ( 1 __ 5 ) (6) x ,y = 7.2 for x =2 and y = 43.2 for

x = 3, hence y = ( 1 __ 5 ) (6) x does not generate 38.4.

For y = 4.8 (2) x , y = 38.4 for x = 3; ans. y = 4.8 (2) x

36a. f(x) = 20 (1.2) x f(2) = 20( 1.2) 2 f(2) = 28.8; $28.80

b. f(x) = 20 (1.2) x 100 ≥ 20 (1.2) x x ≥ 9; after 9 weeks


c. f(x) = 20 (1.2) x f(0) = 20 (1.2) 0 f(0) = 20; $20

d. increase = f(n + 1)

_ f(n)

- 1

increase = 20 (1.2) n + 1

__________ 20 (1.2) n

- 1

increase = .2; .2 or 20%

37. If the value of b were 1, the function would be constant. If the value of a were 0, the function would be the constant function y = 0.

38. Possible answer: The graphs have the same basic shape and the same y-intercept; each graph is steeper than the one before it.

39. Possible answer: The graphs have the same basic shape and the same y-intercept; each graph is steeper than the one before it.

40. f(x) = 4 x f(3) = 4 3 f(3) = 64

41. f(x) = - (0.25) x f(1.5) = - (0.25) 1.5 f(1.5) = -0.125

42. f(x) = 0.4 (10) x f(-3) = 0.4 (10) -3 f(-3) = 0.0004 or 4 × 10 -4

43a. In 2001, n = 0C = 2000 (1.08) 0 C = 2000; $2000

b. increase = 2000 (1.08) n + 1

____________ 2000 (1.08) n

- 1

increase = 0.08; 8%

c. For 2006, n = 5C = 2000 (1.08) 5 C = 2938.66; $2938.66

44. Possible answer: The following table shows how much money you could earn with each plan.

YearSalary Plan A

Salary Plan B

0 $0 $10,000

1 $20,000 $20,000

2 $40,000 $40,000

3 $60,000 $80,000

Choose plan B because plan A doesn’t pay anything for the first year and because after 3 years, plan B pays more money.

45. C; the other graphs do not increase exponentially.

46. G; f(4) = 15 (1.4) 2 = 29.4

47. D; a 1 = 5, r = 5, hence a n = 5 (5) n - 1 = 5 n


48. 4 x = 64 4 x = 4 3 x = 3

49. ( 1 __ 3 )

x = 1 ___

27

3 -x = 3 -3 -x = -3 x = 3

50. 2 x = 1 ___ 16

2 x = 1 ___ 24

2 x = 2 -4 x = -4

51. The value of a is the y-intercept.


52. 88 + 89 + x __________ 3 ≥ 90

x ≥ 93

53. 25; x 2 + 10x + 25 = (x + 5) 2

54. 32x; 4 x 2 + 32x + 64 = (2x + 8) 2

55. 9 x 2 ; 9 x 2 + 42x + 49 = (3x + 7) 2

56. a n = 4 (3) n - 1 a 12 = 4 (3) 11 a 12 = 708,588


CONNECTING ALGEBRA TO GEOMETRY: CHANGING DIMENSIONS, PAGE 779

TRY THIS, PAGE 779

1. widths: 8, 4, 2, 1; common ratio: 1 __ 2

lengths: 16, 8, 4, 2; common ratio: 1 __ 2

heights: 32, 16, 8, 4; common ratio: 1 __ 2

volumes: 4096, 512, 64, 8; common ratio: 1 __ 8

2. heights: 8, 24, 72; common ratio: 3edge of bases: 3, 9, 27; common ratio: 3volumes: 24, 648, 17,496; common ratio: 27

ALBEGRA LAB, PAGE 780

TRY THIS, PAGE 780

1. doubles

2. 2 0 , 2 1 , 2 2 , 2 3 , 2 4 , 2 5

3. number of regions = 2 (2) n - 1 = 2 n

4. n = 8; number of regions = 2 8 = 256

5. 2 n = 512 2 n = 2 9 n = 9; 9 folds

6. is divided in half

7. 2 0 , 2 -1 , 2 -2 , 2 -3 , 2 -4 , 2 -5

8. a n = 1 __ 2 ( 1 __

2 )

n - 1

a n = ( 1 __ 2 )

n

a n = 2 -n

9. a 7 = 2 -7 = 1 ____ 128

10. 2 -n = 1 ____ 256

2 -n = 1 ___ 28

2 -n = 2 -8 n = 8; 8 cuts

11-3 EXPONENTIAL GROWTH AND DECAY, PAGES 781−788

CHECK IT OUT PAGES 781−784

1. y = a (1 + r) t = 1200 (1.08) t ;In 2006, y = 1200 (1.08) 6 = $1904.25

2a. A = P (1 + r __ n ) nt

= 1200 (1 + 0.035 _____ 4 )

4t

= 120 0(1.00875) 4t; After 4 years, A = 1200 (1.00875) 16 = $1379.49

b. A = P (1 + r __ n ) nt

= 4000 (1 + 0.03 ___ 12

) 12t

= 4000 (1.0025) 12t After 8 years, A = 4000 (1.0025) 96 = $5083.47

3. y = a (1 - r) t = 48,000 (1 - 0.03) t = 48,000(0 .97) t After 7 years, y = 48,000 (0.97) 7 = 38,783

4a. t = 180 years

_________ 30 years

= 6A = P( 0.5) t = 100( 0.5) 6 = 1.5625 mg

b. t = 5 weeks _______ 5 days

= 7A = P(0.5) t = 100 (0.5) 7 = 0.78125 g


1. Possible answers: interest earned on an investment, population growth or decline, radioactive decay

2. increasing; by 2% per year

3. An exponential growth function has the form y = a (1 + r) t . The base (1 + r) corresponds to the base b. The exponent t corresponds to the exponent x. An exponential decay function has the form y = a (1 - r) t . The base (1- r) corresponds to the base b. The exponent t corresponds to the exponent x.

4.


1. exponential growth, since 2 > 1.


2. y = a (1 + r) t = 12,000 (1 + 0.06) t = 12,000 (1.06) t After 4 years, y = 12,000 (1.06) 4 = $15,149.72

3. y = a (1 + r) t = 300 (1 + 0.08) t = 300 (1.08) t After 5 years, y = 300 (1.08) 5 = 441

4. A = P (1 + r __ n ) nt

= 1500 (1 + 0.035 _____ 1 )

t

= 1500 (1.035) t After 4 years, A = 1500 (1.035) 4 = $1721.28

5. A = P (1 + r __ n ) nt

= 4200 (1 + 0.028 _____ 4 )

4t

= 4200 (1.007) 4t Ater 6 years, A = 4200 (1.007) 24 = $4965.43

6. y = a (1 - r) t = 18,000 (1 - 0.12) t = 18,000 (0.88) t After 10 years, y = 18,000 (0.88) 10 = $5013.02

7. y = a (1 - r) t = 10 (1 - 0.16) t = 10 (0.84) t After 4 hours, y = 10 (0.84) 4 = 4.98 mg

8. t = 1 hr _______ 20 min

= 3A = P (0.5) t = 30 (0.5) 3 = 3.75 g

9. t = 156 days

________ 52 days

= 3A = P (0.5) t = 44 (0.5) 3 = 5.5 g


10. y = a (1 + r) t = 149,000 (1.06) t After 7 years, y = 149,000 (1.06) 7 = $224,040.91


12. A = P (1 + r) nt = 70 0 (1 + 0.012) 4t = 700 (1.012) 4t After 2 years, A = 700 (1.012) 4t = $770.09

13. y = P (1 + r) nt = 3 0 (1 + 0.078) 2t = 30 (1.079) 2t After 3 years, y = 30 (1.078) 2t = 47 members

14. A = P (1 + r __ n ) nt

= 28,000 (1 + 0.04) t = 28,000 (1.04) t After 5 years, A = 28,000 (1.04) 5 = $34,066.28

15. A = P (1 + r __ n ) nt

= 7000 (1 + 0.03 ____ 4 )

4t

= 7000 (1.0075) 4t After 10 years, A = 7000 (1.0075) 40 = $9438.44

16. A = P (1 + r __ n ) nt

= 3500 (1 + 0.018 _____ 12

) 12t

= 3500 (1.0015) 12t After 4 years, A = 3500 (1.0015) 48 = $3761.09

17. A = P (1 + r __ n ) nt

= 12,000 (1 + 0.026) t = 12,000 (1.026) t After 15 years, A = 12,000 (1.026) 15 = $17,635.66

18. y = a (1 - r) t = 18,000 (1 - 0.02) t = 18,000 (0.98) t After 6 years, y = 18,000 (0.98) 6 = 15,945

19. y = a (1 - r) t = 58 (1 - 0.1) t = 58( 0.9) t After 8 years, y = 58 (0.9) 8 = $24.97

20. t = 6 days

________ 36 hours

= 144 hours _________ 36 hours

= 4A = P(0.5) t = 80 (0.5) 4 = 5 g

21. growth; 61%, since 1+ r = 1.61

22. decay; 90.2%, since 1 - r = 0.098

23. decay; 33 1 __ 3 %, since 1 - r = 2 __

3

24. growth; 50%, since 1 + r = 3 __ 2

25. growth; 10%, since 1 + r = 1.1

26. decay; 20%, since 1 - r = 0.8

27. growth; 25%, since 1 + r = 5 __ 4

28. decay; 50%, since 1 - r = 1 __ 2

29. y = a (1 + r) t = 58,000,000 (1.001) t After 3 years, y = 58,000,000 (1.001) 3 = 58,174,174

30. y = a (1 + r) t = 32,000 (1.07) t After 5 years, y = 32,000 (1.07) 5 = $44,881.66


31. y = a (1 - r) t = 8200 (1 - 0.02) t = 8200 (0.98) t After 7 years, y = 8200 (0.98) 7 = $7118.63

32. y = a (1 - r) t = 25,000 (1 - 0.15) t = 25,000 (0.85) t After 6 years, y = 25,000 (0.85) 6 = $9428.74


34. t = 3500 years

__________ 5700 years

= 35 ___ 57

A = P (0.5) t

= 15 (0.5) 35 ___ 57

≈ 9.8 g

35. B; possible answer: student B did not subtract the rate from 1.

36. No; possible answer; there is no value for t that would make (0.84) t equal 0.

37. y = a (1 + r) t 600 = 300( 1 + 0.04) t 2 = 1.04 t t ≈ 18 years

38a. y = a (1 + r) r = 20,000 (1.09) t

b. In 2008, t = 6, hencey = 20,000 (1.09) 6 = $33,542

c. 2011

Year Tuition ($)

2002 20,000

2003 21,800

2004 23,762

2005 25,900.58

2006 28,231.63

2007 30,772.48

2008 33,542.00

2009 36,560.78

2010 39,851.25

2011 43,437.87

39. In 10 years:A: 600 (1.05) 10 = $977.34

B: 500 (1 + 0.06 ____ 4 )

40 = 500 (1.015) 40

= $907.01A will have a larger balance.In 20 years:A: 600 (1.05) 20 = $1591.98B: 500 (1.015) 80 = $1645.33B will have a larger balance.

40. 50 h; 15h

41. The graph when r is 20% rises faster than when r is 10%. The greater the value of r, the faster the graph will rise.

42. Possible answer: $400 is invested at a rate of 8% compounded annually.

43. Possible answer: The population is 800 and decreasing at a rate of 4% per year.

44. No; possible answer: the sample doubles every minute, so the container is half full 1 minute before it is full. This would be after 5 min.

45. D; y = a (1 -r) t a = 500, 1 - r = 1 - 0.01 = 0.99

46. G; a = -5, so as the absolute value of y decreases, y is actually increasing.

47. D; 865 (1.05) 3 = $1001.35

48a. y = a (1 + r) t = 1000 (1 + 0.05) t = 1000 (1.05) t

b. 1300 = 1000 (1.05) t t ≈ 5; about 2005


49. about 20 years 50. y = a (1 + r) t 1000 = 500 (1.04) t t ≈ 18 yr for r = 0.081000 = 500 (1.08) t t ≈ 9 yr

51. A = P (0.5) t 10 = 80 (0.5) t t = 3So, half-life = 300 ____

t = 100 min or 1 h 40 min

52. A = P (0.5) t

15 = P (0.5) 6 __ 2

P = 120 g

53. A = P (1 + r __ n ) nt

250,000 = P (1 + 0.013) (4 · 8) P = $225,344

54. Month Balance ($)Monthly

Payment ($)Remaining Balance ($)

1.5% Finance

Charge ($)

New Balance ($)

1 200 30 170 2.55 172.55

2 172.55 30 142.55 2.14 144.69

3 144.69 30 114.69 1.72 116.41


4 116.41 30 86.41 1.30 87.71

5 87.71 30 57.71 0.87 58.58

6 58.58 30 28.58 0.43 29.01

7 29.01 29.01 0 0 0

b. Table shows balance is paid off in 7 months.

c. (6(30) + 29.01) - 200 = 9.01


55. 1.2 ___ 1.5

= h ___ 20

h = 16 ft

56. w ___ 12

= 10 ___ 20

w = 6 in.

57. f(x) = 2x + 1 58. f(x) = x - 4

59. f(x) = x 2 -1 60. f(4) = 0.10 (2) 4 = $1.60;12.80 = 0.10 (3) x x = 7 days

11-4 LINEAR, QUADRATIC AND EXPONENTIAL MODELS, PAGES 789−795


1a. exponential b. quadratic

2. Quadratic; for every constant change in the x-values of +1, there is a constant second difference of -6 in the y-values.

3. The oven temperature decreases by 50°F every 10 minutes; y = -5x + 375; 75°F

THINK AND DISCUSS

1. No; most real-world data probably will not fit exactly into one of these patterns.

2. No; this is just a prediction based on the assumption that the observed trends will continue, which they may or may not do.

3.


1. exponential 2. quadratic

3. linear 4. Quadratic; for every constant change of +1 in the x-values, there is a constant second difference of -1 in the y-values.

5. Exponential; for every constant change of +1 in the x-values, there is a constant ratio of 2.

6. Linear; for every constant change of +1 in the x-values, there is a constant change of +2 in the y-values.

7. Grapes cost $1.79/lb; y = 1.79x; $10.74

PRACTICE AND PROBLEM SOLVING. PAGES 793−795

8. quadratic 9. linear


10. exponential 11. Linear, for every constant change of +1 in the x-values, there is a constant change of -1 in the y-values.

12. Quadratic, for every constant change of +1 in the x-values, there is a constant second difference of -2 in the y-values.

13. Exponential, for every constant change of +1 in the x-values, there is a constant ratio of 0.5 in the y-values.

14. The company’s sales are increasing by 20% each year; y = 25,000 (1.2) x ; $154,793.41

15. l = 6k; linear with m = 6 and b = 0

16. Linear; for every weekly interval, the height of the plant has a constant increase of 0.5 inches.

17. Linear; for each successive year, the number of games has a constant change of 0.

18. Quadratic; for each successive time interval, the height of a ball has a constant second difference of -0.28.

19. y = 0.2 (4) x 20. y = - 1 __ 2 x + 4

21. linear 22. quadratic

23. Possible answer: (0,3), (1,6), (2,12), (3,24); for a constant change in x of +1, there is a common ratio of 2.

24. Possible answer: the first differences are constant, so there is no need to check the second differences. A linear function would best model the data.

25. Possible answer: make a table of ordered pairs and see whether the y-values show a pattern of constant second differences or constant ratios.

26a. college 1: linear because it has constant changes of $200 each year; college 2: exponential because it has a constant yearly ratio of 1:1.1.

b. college 1: y = 200x + 2000; college 2: y = 2000 (1.1) x

c. Both have the same tuition ($2000) in 2004.

d. For college 1, $200 is added each year, so 2000 + 200 = 2200. For college 2, 10% is added each year, so 2000 + (0.1)(2000) = 2200.

27. C; the data is linear since it has a constant change in the y-values for each constant change in the x-values.

28. F; 2% is a common ratio.

29. C; For every constant change of +1 in the x-values, there is a constant change of +2 in the y-values.


30a. Year Value ($)

0 18,000

1 15,120

2 12,700.80

3 10,668.67

4 8961.68

Year 0 is the year when the car is purchased.

b. exponential, for each successive year, the value decreases by 16%, the common ratio.

c. y = 18,000 (0.84) x

d. y = 18,000 (0.84) 5 1 __

2 = $6899.36

e. y = 18,000 (0.84) 8 = $4461.77

31a. Possible answer: quadratic; the second differences are approximately constant at -2.

b. about 48 kg

c. No; this quadratic model will begin to decrease although the dog’s weight will either continue to grow or eventually remain constant.


32. 5n; she would run 5 km n times.

33. 145 ____ g 34. 74 - b

35. 4 x 2 = 100 x 2 = 25 x = ± √ �� 25 x = ±5

36. 10 - x 2 = 10 - x 2 = 0 x = 0

37. 16 x 2 + 5 = 86 16 x 2 = 81

x 2 = 81 ___ 16

x = ± √ �� 81 ___ 16

x = ± 9 __ 4

38. y = 6 x

39. y = -2 (5) x 40. y = ( 1 __ 3 )

x

MULTI-STEP TEST PREP, PAGE 796

1. y = 350 (1.09) x where y = tuition is the dependent variable and x = years since 1980 is the independent variable.


2. y = 350 (1.09) 26 = $3289.71

3. Answers will vary.

4. 700 = 350 (1.09) x x ≈ 8; about 1988

å

5. 1000 = 350 (1.09) x about 1992-1993

READY TO GO ON? PAGE 797

1. 6 __ 3 = 2; 12 ___

6 = 2; 24 ___

12 = 2; the common ratio is 2

next 3 terms: 24(2) = 48, 48(2) = 96, and 96(2) = 192

2. 2 ___ -1 = -2; -4 ___

2 = -2; 8 ___ -4

= -2;

the common ratio is -2next 3 terms: 8(-2) = -16, (-16)(-2) = 32, and 32(-2) = -64

3. -1200 ______ -2400 = 1 __

2 ; -600 ______ -1200

= 1 __ 2 ; -300 _____ -600

= 1 __ 2

next 3 terms: -300 ( 1 __ 2 ) = -150, -150 ( 1 __

2 ) = -75,

and -75 ( 1 __ 2 ) = -37.5

4. a n = a 1 r n-1 a 8 = (2) (3) 8-1 a 8 = 4374

5. a 1 = 1000, r = 4 __ 5

a n = a 1 r n-1

a 7 = 1000 ( 4 __ 5

) 7-1

a 7 = 262.144 cm

6. f(x) = 3 (1.1) x

f(4) = 3 (1.1) 4 f(4) = 4.39 in

7. y = 3 x

8. y = 2( 2) x 9. y = -2 (4) x

10. y = - (0.5) x 11. f(x) = 40 (0.8) x

2 = 40 (0.8) x x ≈ 14; after about 14 h

12. y = a (1 + r) x = 30,000 (1.03) x; After 10 years, y = $40,317.49

13. y = a (1 + r __ n ) nx

= 2000 (1.00375) 12x; After 3 years, y = $2288.50

14. y = a( 1 - r) x = 1200 (0.8) x After 4 years, y = $491.52


15. A = P (0.5) t

A = 100(0. 5) 300 ____ 30

A = 100(0. 5) 10

A = 0.098 mg

16. quadratic 17. exponential

18. linear; for every constant change of +1 in the x-values, there is a constant change of +1 in the y-values.

19. exponential: for every constant change of +1 in the x-values, there is a common ratio of 1 __

2 in the

y-values.

20. The value of the stamp is increasing by 20% each year; y = 5 (1.2) x ; $37.15

11-5 SQUARE-ROOT FUNCTIONS, PAGES 798−803


1a. y = 8 √ � x = 8 √ �� 25 = 40.00 ft/s

b. y = 8 √ � x = 8 √ �� 15 = 30.98 ft/s

2a. y = √ �� 2x - 1 2x -1 ≥ 0 2x ≥ 1

x ≥ 1 __ 2

Domain: { x | x ≥ 1 __ 2 }

b. y = √ �� 3x - 5 3x - 5 ≥ 0 3x ≥ 5

x ≥ 5 __ 3

Domain: { x | x ≥ 5 __ 3 }

3a. f(x) = √ � x + 2 b. f(x) = 2 √ � x + 3


1. Possible answer: Set the expression under the square-root sign greater than or equal to zero and solve.

2. The graph of f(x) = √ �� x + 8 is the graph of f(x) = √ � x translated 8 units to the left.

3. The graph of f(x) = √ �� x + 8 is the graph of f(x) = √ � x translated 8 units to the left, while the graph of f(x) = √ � x + 8 is the graph of f(x) = √ � x translated 8 units up.

4.


1. There is no variable under the square-root sign.

2. c = √ �� a 2 + b 2

= √ �� 14 2 + 8 2 = 16.12 cm

3. y = √ �� x + 6 x + 6 ≥ 0 x ≥ -6Domain:{x | x ≥ -6}

4. y = 4 - √ �� 3 - x 3 - x ≥ 0 -x ≥ -3 x ≤ 3Domain:{x | x ≤ 3}

5. y = √ � 2x - 52x ≥ 0 x ≥ 0Domain: {x | x ≥ 0}

6. y = √ �� x +2 x + 2 ≥ 0 x ≥ -2Domain:{x | x ≥ -2}

7. y = √ �� 3x +9 3x + 9 ≥ 0

x ≥ - 9 __ 3

x ≥ -3Domain:{x | x ≥ -3}

8. y = x + √ �� x - 5 x - 5 ≥ 0 x ≥ 5Domain:{x | x ≥ 5}

9. f(x) = √ �� x - 1 10. f(x) = - √ � 2x


11. f(x) = √ � x + 1 12. f(x) = √ � x - 12

13. f(x) = √ �� 4 - x 14. f(x) = √ �� x +4

15. f(x) = √ �� 24x f(104) = √ �� 24 · 104 f(104) ≈ 49.96 mi/h

16. y = √ �� 8 - 2x 8 - 2x ≥ 0 -2x ≥ -8 x ≤ 4Domain: { x | x ≤ 4 }

17. y = 4 - √ � x __ 2

x __ 2 ≥ 0

x ≥ 0Domain: { x | x ≥ 0 }

18. y = √ �� 3x + 2 3x + 2 ≥ 0

x ≥ - 2 __ 3

Domain: { x | x ≥ - 2 __

3 }

19. y = √ �� -2x + 3 -2x + 3 ≥ 0 -2x ≥ -3

x ≤ 3 __ 2

Domain: { x | x ≤ 3 __

2 }

20. y = 2 √ �� x + 1 x + 1 ≥ 0 x ≥ -1Domain: { x | x ≥ -1 }

21. y = √ �� 3(x + 2) - 1 3(x + 2) - 1 ≥ 0

x + 2 ≥ 1 __ 3

x ≥ - 5 __ 3

Domain: { x | x ≥ - 5 __

3 }

22. y = √ �� 2(x + 4) - 32(x + 4) ≥ 0 x + 4 ≥ 0 x ≥ -4Domain: { x | x ≥ -4 }

23. y = 7 √ �� x __ 5 - 8

x __ 5 - 8 ≥ 0

x ≥ 40Domain: { x | x ≥ 40 }

24. y = √ �� 2(3x - 6) 2(3x - 6) ≥ 0 3x -6 ≥ 0 x ≥ 2Domain: { x | x ≥ 2 }

25. y = √ �� 1 __ 3 (x - 9)

1 __ 3 (x - 9) ≥ 0

x - 9 ≥ 0 x ≥ 9Domain: { x | x ≥ 9 }

26. y = √ �� 2(x + 7) - 6 2(x + 7) - 6 ≥ 0 x + 7 ≥ 3 x ≥ -4Domain: { x | x ≥ -4 }

27. y = 4 + √ �� 3x + 2 3x + 2 ≥ 0

x ≥ - 2 __ 3

Domain: { x | x ≥ - 2 __

3 }

28. f(x) = √ �� x - 5 29. f(x) = √ � 2x - 4

30. f(x) = -1 - √ � x 31. f(x) = √ � x - 4

32. f(x) = 3 √ �� x - 6 33. f(x) = 1 __ 2 √ �� x + 4

34. r = √ � A __ π

= √ �� 60 ____ 3.14

= 4.37 cm

35a.

b. For each function, √ � x must be real, hencex ≥ 0Domain: { x | x ≥ 0 }

c. √ � x ≥ 0 for all values of x in the domain.Range: { y | y ≥ 0 }

d. Possible answer: it has a minimum value of 0 and curves to the right. As a increases, the curve becomes steeper.

36a.


b. For each function, √ � x must be real, hencex ≥ 0Domain: { x | x ≥ 0 }

c. √ � x ≥ 0 for all values of x in the domain and the coefficients in all the functions are negative.Range: { y | y ≤ 0 }

d. Possible answer: it has a maximum value of 0 and curves to the right. As a decreases, the curve becomes steeper.

37. d = √ �� (w - x) 2 + (z - y) 2

= √ �� (5 - 2) 2 + (3 - 1) 2

= √ �� 13 = 3.61 units

38.

f(x) = √ �� 9.8x f(500) = √ �� 9.8 · 500 f(500) = 70 m/s

39. v = √ �� 2gr Mercury:

v = √ �� 2 · 3.7 · 2.4 · 10 6 = 4214 m/sVenus:

v = √ �� 2 · 8.8 · 6.1 · 10 6 = 10,361 m/sEarth:

v = √ �� 2 · 9.8 · 6.4 · 10 6 = 11,200 m/sMars:

v = √ �� 2 · 3.7 · 3.4 · 10 6 = 5016 m/s

40. V = π r 2 h

r = √ �� V ___ πh

= √ �� 1212 ________ 3.14 · 10

= 6.21 in

41. Set the expression under the square-root sign greater or equal to 0 and solve; the square root of a negative number is not a real number so the domain cannot be all real numbers.

42. Since the domain is x ≥ 5, the value of y is 0 when x = 5.

43. No; the domain of a square-root function is limited to values that make the value under the square-root sign non-negative. A function with a limited domain cannot have a range of all real numbers.

44a. T = 2π √ ��

___ 32

�

___ 32

≥ 0

� ≥ 0Domain: { � | � ≥ 0 }

b. T = 2π √ ��

___ 32

= 2 · 3.14 √ �� 80 ___ 32

= 9.93 s

c. No; 9.93 seconds is too fast for the ride to make one complete swing back and forth. This is for a pendulum that is under the influence of gravity only. This is not true for the ride.

45. A; the graph of √ � x is shifted 3 units left.

46. J; x ≥ 2 would make √ �� x - 2 a nonnegative number

47. C; y = √ �� 1 __ 5 · 25 = 2.2 seconds

48. g(x) = √ � 4x - 1g(9) = √ �� 4(9) - 1g(9) = 5


49. y = √ �� x 2 - 25 x 2 - 25 ≥ 0 x 2 ≥ 25 |x| ≥ 5Domain: { x | x ≤ -5 or x ≥ 5 }

50. y = √ �� x 2 + 5x + 6 x 2 + 5x + 6 ≥ 0(x + 2)(x + 3) ≥ 0x + 2 ≥ 0 and x + 3 ≥ 0or x + 2 ≤ 0 and x + 3 ≤ 0Domain: { x | x ≤ -3 or x ≥ -2 }

51. y = √ �� 2 x 2 + 5x - 12 2 x 2 + 5x -12 ≥ 0(2x - 3)(x + 4) ≥ 02x - 3 ≥ 0 and x + 4 ≥ 0or 2x - 3 ≤ 0 and x + 4 ≤ 0

Domain: { x | x ≥ 3 __ 2 or x ≤ -4 }

52. y = 2 - √ �� x + 3 x + 3 ≥ 0x ≥ -3 and y ≤ 2Domain: { x | x ≥ -3 }Range: { y | y ≤ 2 }

53. y = 4 - √ �� 3 - x 3 - x ≥ 03 ≥ x and y ≤ 4Domain: { x | x ≤ 3 }Range: { y | y ≤ 4 }

54. y = 6 - √ � x __ 2

x __ 2 ≥ 0

x ≥ 0 and y ≤ 6Domain: { x | x ≥ 0 }Range: { y | y ≤ 6 }

55. Possible answers: y = √ � x + b, where b > 0 Example: y = √ � x + 6


56. Possible answers: y = - √ �� x + a + b, where a ≤ 0 and b < 0. Example: y = - √ �� x - 1 - 1

57a. 2, 4; when x = 2 or x = 4, the expression under the square-root sign is negative.

b. for x = 5, y = 3 - √ �� 2(5 - 5) = 3for x = 7, y = 3 - √ �� 2(7 - 5)

= 3 - 2 = 1


58. 2y = 4x - 8 y = 2x - 4

59. 3x + 6y = 12 6y = -3x + 12

y = - 1 __ 2 x + 2

60. 2x = -y - 9 y = -2x - 9

61. (a + b) 2 = a 2 + 2ab + b 2 (3x - 1) 2 = (3x) 2 + 2(3x)(-1) + (-1) 2 = 9 x 2 - 6x + 1

62. (a - b)(a + b) = a 2 - b 2 (2x - 5)(2x + 5) = (2x) 2 - (5) 2 = 4 x 2 - 25

63. (a + b) 2 = a 2 + 2ab + b 2

(a - b 2 c) 2 = a 2 + 2(a)(- b 2 c) + (- b 2 c) 2 = a 2 - 2a b 2 c + b 4 c 2

64. (a + b) 2 = a 2 + 2ab + b 2

( x 2 + 2y) 2 = ( x 2 ) 2 + 2( x 2 )(2y) + (2y) 2 = x 4 + 4 x 2 y + 4 y 2

65. (a - b)(a + b) = a 2 - b 2 (3r - 2s)(3r + 2s) = (3r) 2 - (2s) 2 = 9 r 2 - 4 s 2

66. (a - b)(a + b) = a 2 - b 2

( a 3 b 2 - c 4 )( a 3 b 2 + c 4 ) = ( a 3 b 2 ) 2 - ( c 4 ) 2 = a 6 b 4 - c 8

67. A = P (1 + r __ n ) nt

A = 42,000 (1 + 0.05 ____ 4 )

4t

= 42,000 (1.0125) 4t After 3 years, A = 42,000 (1.0125) 12 = $48,751.69

68. A = P (0.5) t

t = 1 day _________

3.25 hours = 24 hours _________

3.25 hours = 96 ___

13

A = 230 (0.5) 96 ___ 13

= 1.38 g

TECHNOLOGY LAB: GRAPH RADICAL FUNCTIONS, PAGE 804

TRY THIS, PAGE 804

1.

2.

3. The graph of f(x) = √ �� x + 1 + 4 will be the graph of f(x) = √ � x shifted 1 unit left and 4 units up.

4. The graph of f(x) = 2 √ � x will have a steeper curve.

11-6 RADICAL EXPRESSIONS, PAGES 805–810

CHECK IT OUT! PAGES 805–807

1a. √ �� 256 ____ 4 = √ �� 64

= 8

b. √ �� 40 + 9 = √ �� 49 = 7

c. √ �� 5 2 + 12 2 = √ �� 25 + 144

= √ �� 169 =13

d. √ �� (3 - x) 2 = 3 - x


2a. √ �� 128 = √ �� 64(2)

= √ �� 64 √ � 2

= 8 √ � 2

b. √ �� x 3 y 2 = √ � x 3 √ � y 2

= √ � x √ � x 2 √ � y 2 = xy √ � x

c. √ �� 48 a 2 b = √ �� 16 √ � 3 √ � a 2 √ � b = 4a √ �� 3b

3a. √ �� 12 ___ 27

= √ � 4 __ 9

= √ � 4 ___ √ � 9

= 2 __ 3

b. √ �� 36 ___ x 4

= √ �� 36

____ √ � x 4

= 6 __ x 2

c. √

��

y 6 __

4 =

√ � y 6 ____

√ � 4

= y 3 ___

2

4a. √ �� 20 ___ 49

= √ �� 20

____ √ �� 49

= √ � 4 √ � 5

______ √ �� 49

= 2 √ � 5

____ 7

b. √ ��

z 5 ____ 25 y 2

= √ � z 5 ________ √ �� 25 √ � y 2

= √ �� (z) z 4

______ 5y

= z 2 √ � z _____ 5y

c. √ ��

p 6

___ q 10

= √ � p 6

_____ √ �� q 10

= p 3

__ q 5

5. c = √ �� a 2 + b 2

= √ �� 60 2 + 60 2

= √ �� (2) 60 2 = 60 √ � 2 ft or 84.9 ft


1. Method 1: √ �� 16(9) = √ �� 144 = 12Method 2: √ �� 16(9) = √ �� 16 √ � 9 = 4(3) = 12

2. Method 1: √ �� 100 ____ 4 = √ �� 25

= 5

Method 2: √ �� 100 ____ 4 =

√ �� 100 _____

√ � 4

= 10 ___ 2

= 5

3.


1. 3x - 6 is the radicand 2. √ �� 81 = 9

3. √ �� 98 ___ 2 = √ �� 49

= 7

4. √ �� (a + 7) 2 = a + 7

5. √ �� 180 = √ �� (36)5 = √ �� 36 √ � 5 = 6 √ � 5

6. √ �� 40 = √ �� (4)10 = √ � 4 √ �� 10 = 2 √ �� 10

7. √ �� 648 = √ �� (324)2 = 18 √ � 2

8. √ �� m 5 n 3 = √ �� m 5 √ � n 3

= √ �� m 4 √ � n 2 √ �� mn = m 2 n √ �� mn

9. √ �� 32 x 4 y 3 = √ �� 16(2) x 4 y 2 y = 4 x 2 y √ � 2y

10. √ �� 200 a 2 b = √ �� (2)100 a 2 b

= √ �� 100 a 2 √ �� 2b = 10a √ �� 2b

11. √ �� 17 ___ 25

= √ �� 17 ____ √ �� 25

= √ �� 17 ____

5

12. √ �� 7 ___ 16

= √ � 7 ____

√ �� 16

= √ � 7 ___ 4

13. √ �� 6 ___ 49

= √ � 6

____ √ �� 49

= √ � 6

___ 7

14. √ �� b __ c 2

= √ � b

____ √ � c 2

= √ � b

___ c

15. √ ��

4 x 2 ____ 36x

= √ � x __ 9

= √ � x ___

√ � 9

= √ � x ___

3

16. √ ��

7 a 4 ___ 9 a 3

= √ �� 7a ___ 9

= √ �� 7a ____

√ � 9

= √ �� 7a ____

3

17. √ �� 108 ____ 49

= √ �� (36)3

_______ √ �� 49

= √ �� 36 √ � 3

_______ 7

= 6 √ � 3

____ 7

18. √ �� 204 ____ 25

= √ �� (4)51

______ √ �� 25

= √ � 4 √ �� 51

_______ 5

= 2 √ �� 51

_____ 5


19. √ �� 512 ____ 81

= √ �� 256(2)

_______ √ �� 81

= √ �� 256 √ � 2

________ √ �� 81

= 16 √ � 2 _____ 9

20. √ �� 1 ____ 36 x 2

= √ � 1 ________ √ �� 36 √ � x 2

= 1 ___ 6x

21. √ ��

50 x 2 ____ 169

= √ �� (25)2 x 2

________ √ �� 169

= √ �� 25 √ � 2 √ � x 2

__________ √ �� 169

= 5x √ � 2 _____ 13

22. √ ��

72 x 7 ____ 4 x 4

= √ �� 18 x 3

= √ � 9 √ � x 2 √ � 2x = 3x √ � 2x

23. c = √ �� a 2 + b 2

= √ �� 20 2 + 25 2 = √ �� 25(16 + 25) = √ �� 25 √ �� 41 = 5 √ �� 41 mi ≈ 32 mi


24. √ �� 100 = 10 25. √ �� 800 ____ 2 = √ �� 400

= 20

26. √ �� 3 2 + 4 2 = √ �� 25 = 5

27. √ �� 3 · 27 = √ �� 81 = 9

28. √ � a 4 = a 2 29. √ �� (x + 1) 2 = x + 1

30. √ �� (5 - x) 2 = 5 - x 31. √ �� (x - 3) 2 = x - 3

32. √ �� 125 = √ �� 25(5) = √ �� 25 √ � 5 = 5 √ � 5

33. √ �� 4000 = √ �� 10(400) = 20 √ �� 10

34. √ �� 216 a 2 b 2 = √ �� 36 a 2 b 2 √ � 6 = 6ab √ � 6

35. √ �� 320 r 2 s 2 = √ �� (5)64 r 2 s 2 = 8rs √ � 5

36. √ �� 15 ___ 64

= √ �� 15

____ √ �� 64

= √ �� 15

____ 8

37. √ �� 45 ___ 4 =

√ �� 9(5) _____

√ � 4

= √ � 9 √ � 5

______ 2

= 3 √ � 5

____ 2

38. √ ��

64 a 4 ____ 4 a 6

= √ �� 16 ___ a 2

= √ �� 16

____ √ � a 2

= 4 __ a

39. √ ��

14 z 3 ____ 9 z 3

= √ �� 14 ___ 9

= √ �� 14 ____ √ � 9

= √ �� 14 ____

3

40. √ �� 128 ____ 81

= √ �� 128

_____ √ �� 81

= √ �� 64(2)

______ √ �� 81

= 8 √ � 2 ____ 9

41. √ ��

x 3 __ y 6

= √ �� x 2 x _____ √ � y 6

= x √ � x ____ y 3

42. √ �� 150 _____ 196 x 2

= √ �� 6(25)

_______ √ �� 196 x 2

= √ �� 25 √ � 6

_______ 14x

= 5 √ � 6

____ 14x

43. √ ��

192 s 3 _____ 49s

= √

��

64(3) s 2 _______

49

= √ �� 64 s 2 √ � 3

_________ √ �� 49

= 8s √ � 3

_____ 7

44. t = √ �� d ___ 16

= √ �� 160 ____ 16

= √ �� 10 s; 3.2 s

45. -4 √ �� 75 = -4 √ �� 25(3)

= -4 √ �� 25 √ � 3

= -20 √ � 3

46. - √ �� 80 = - √ �� 16(5)

= - √ �� 16 √ � 5

= -4 √ � 5

47. 5x √ �� 63 = 5x √ �� 9(7)

= 5x √ � 9 √ � 7

= 15x √ � 7

48. 3 √ �� 48x = 3 √ �� 16(3)x

= 3 √ �� 16 √ � 3x

= 12 √ � 3x

49. 2 √ ��

x 2 __ 4 = 2

√ � x 2 ____ √ � 4

= (2) x __ 2

= x

50. 1 __ 2 √ �� 1 ___

25 = 1 __

2

√ � 1 ____ √ �� 25

= 1 __ 2 ( 1 __

5 )

= 1 ___ 10

51. 3x √ ��

x 5 ___ 81

= 3x √ �� x 4 x _____ √ �� 81

= 3x √ � x 4 √ � x ________ 9

= 3 x 3 √ � x ______ 9

= x 3 √ � x _____ 3

52. 12 ___ x √

��

x 2 y ___

36 = 12 ___ x

√ � x 2 √ � y _______

√ �� 36

= ( 12 ___ x ) x √ � y

____ 6

= 2 √ � y

53. √ �� 12 √ � 3 = √ �� 12(3)

= √ �� 36 = 6

54. √ �� 18 √ � 8 = √ �� 18(8) = √ �� 144 = 12

55. √ �� 10 √ � 5 = √ �� 10(5)

= √ �� 50 = √ � 2 √ �� 25 = 5 √ � 2

56. √ � 8 √ �� 14 = √ �� 8(14)

= √ �� 112

= √ �� 16(7)

= 4 √ � 7

57. √ �� 33

____ √ � 11

= √ �� 33 ___ 11

= √ � 3


58. √ �� 24 ____ √ � 2

= √ �� 24 ___ 2

= √ �� 12 = √ �� 3(4)

= 2 √ � 3

59. √ �� 60

____ √ � 3

= √ �� 60 ___ 3

= √ �� 20

= √ �� 4(5)

= 2 √ � 5

60. √ �� 72 ____ √ � 9

= √ �� 72 ___ 9

= √ � 8

= √ �� 4(2)

= 2 √ � 2

61. 42 ft;length of missing side = √ �� 10 2 + 14 2 ≈ 17.2 ft10 + 14 + 17.2 = 41.2 ft, rounded up to 42 ft.

62. Possible answer: Use the Quotient Property of Square Roots:

√ �� 28 ___ 49

= √ �� 28

____ √ �� 49

Then use the Product Property of Square Roots in the numerator:

√ �� 28

____ √ �� 49

= √ �� 4 · 7 ______ √ �� 49

= √ � 4 √ � 7 ______ √ �� 49

Then simplify by taking the square roots of the perfect squares:

√ � 4 √ � 7 ______ √ �� 49

= 2 √ � 7 ____ 7

63a. v = √ �� 64h = √ �� 64 √ � h = 8 √ � h ;v = 8 √ �� 137 ≈ 93.6 ft/s

b. Pythagorean Theorem

c. d = √ �� x 2 + h 2

= √ �� 103 2 + 137 2 = 171.4 ft

64. Possible answer: The square root of a negative number is not a real number.

65. d = √ �� 6h

____ 3

Sears: d = √ �� 6 · 1450

_________ 3

= √ �� 8700

______ 3

= √ �� 100 √ �� 87

_________ 3

= 10 √ �� 87

______ 3 mi; 31.1 mi

Empire: d = √ �� 6 · 1250

_________ 3

= √ �� 7500

______ 3

= √ �� 2500 √ � 3

_________ 3

= 50 √ � 3

_____ 3 mi; 28.9 mi

Aon: d = √ �� 6 · 1136

_________ 3

= √ �� 6816

______ 3

= √ �� 16 √ �� 426

_________ 3

= 4 √ �� 426

______ 3 mi; 27.5 mi

66. s = 1 __ 2 (a + b + c)

= 1 __ 2 (7 + 9 + 12)

= 14A = √ �� s(s - a)(s - b)(s - c) = √ �� 14(14 - 7)(14 - 9)(14 - 12)

= √ �� 14 · 7 · 5 · 2

= √ �� 14 2 √ � 5 = 14 √ � 5 m 2 ; 31.3 m 2

67. C: 35 is not divisible by a perfect square.

68. F: 2 √ �� 15 = √ � 4 √ �� 15 = √ �� 4 · 15 = √ �� 60

69. C: √ �� 10 2 + 10 2 = √ �� 2( 10 2 ) = 10 √ � 2


70. √ �� 4x + 16 = √ �� 4(x + 4) = √ � 4 √ �� x + 4 = 2 √ �� x + 4

71. √ �� x 3 + x 2 = √ �� x 2 (x + 1)

= √ � x 2 √ �� x+1 = x √ �� x + 1

72. √ �� 9 x 3 - 18 x 2 = √ �� 9 x 2 (x - 2)

= √ �� 9 x 2 √ �� x - 2 = 3x √ �� x - 2

73a. √ � x 2 = x b. √ � x 4 = x 2

c. √ � x 6 = x 3 d. √ � x 8 = x 4

e. √ �� x 10 = x 5


f. x n (since any number to an even power is always positive); x n (since any negative number to an odd power is always negative)


74. yes; possible answer: the equation is y = 6x and is of form y = kx, with k = 6

75. no; possible answer: the equation is y = x - 8 which is not of y = kx form and the

y _ x value is not

the same for each (x, y).

76. y = mx + b

m = -5 - 1 _______ 2-3

= -6 ___ -1

= 6y = 6x + bFor (3, 1), 1 = 6(3) + b b = - 17Hence, y = 6x - 17

77.

exponential

78.

quadratic

11-7 ADDING AND SUBTRACTING RADICAL EXPRESSIONS, PAGES 811–815


1a. 5 √ � 7 - 6 √ � 7 = - √ � 7 b. 8 √ � 3 - 5 √ � 3 = 3 √ � 3

c. 4 √ � n + 4 √ � n = 8 √ � n

d. √ � 2s - √ � 5s + 9 √ � 5s = √ � 2s + 8 √ � 5s

2a. √ �� 54 + √ �� 24 = √ �� 9(6) + √ �� 4(6) = √ � 9 √ � 6 + √ � 4 √ � 6 = 3 √ � 6 + 2 √ � 6 = 5 √ � 6

b. 4 √ �� 27 - √ �� 18 = 4 √ �� 9(3) - √ �� 9(2) = 4 √ � 9 √ � 3 - √ � 9 √ � 2 = 12 √ � 3 - 3 √ � 2

c. √ �� 12y + √ �� 27y = √ �� (4)3y + √ �� (9)3y = √ � 4 √ � 3y + √ � 9 √ � 3y

= 2 √ � 3y + 3 √ � 3y

= 5 √ � 3y

3. 2(2 √ � b + 3 √ � b ) = 2(5 √ � b ) = 10 √ � b in.


1. Group 1: 2 √ � 6 , √ �� 600 = 10 √ � 6 , √ �� 150 = 5 √ � 6 Group 2: 6 √ � 5 , - √ �� 20 = -2 √ � 5 , √ � 5

2. Possible answer: Without simplifying, you cannot tell which terms are like radicals.

3. Possible answer:


1. Possible answer: any pair of a √ � c and b √ � c where a, b are real numbers and c is nonnegative. Example: 4 √ � 6 and -2 √ � 6

2. 14 √ � 3 - 6 √ � 3 = 8 √ � 3 3. 9 √ � 5 + √ � 5 = 10 √ � 5

4. 6 √ � 2 + 5 √ � 2 - 15 √ � 2 = -4 √ � 2

5. 3 √ � 7 + 5 √ � 2 = 3 √ � 7 + 5 √ � 2

6. 5 √ � a - 9 √ � a = -4 √ � a

7. 9 √ �� 6a + 6 √ �� 5a - 4 √ �� 6a = 5 √ �� 6a + 6 √ �� 5a

8. √ �� 32 - √ � 8 = √ �� 16(2) - √ �� 4(2) = √ �� 16 √ � 2 - √ � 4 √ � 2 = 4 √ � 2 - 2 √ � 2 = 2 √ � 2

9. 4 √ �� 12 + √ �� 75 = 4 √ �� 4(3) + √ �� 25(3) = 4 √ � 4 √ � 3 + √ �� 25 √ � 3 = 8 √ � 3 + 5 √ � 3 = 13 √ � 3

10. 2 √ � 3 + 5 √ �� 12 - √ �� 27 = 2 √ � 3 + 5 √ �� 4(3) - √ �� 9(3) = 2 √ � 3 + 5 √ � 4 √ � 3 - √ � 9 √ � 3 = 2 √ � 3 +10 √ � 3 - 3 √ � 3 = 9 √ � 3

11. √ �� 20x - √ �� 45x = √ �� 4(5x) - √ �� 9(5x) = √ � 4 √ � 5x - √ � 9 √ � 5x = 2 √ � 5x - 3 √ � 5x = - √ � 5x

12. √ �� 28c + 9 √ �� 24c = √ �� 4(7c) + 9 √ �� 4(6c) = √ � 4 √ � 7c + 9 √ � 4 √ � 6c = 2 √ � 7c + 18 √ � 6c

13. √ �� 50t - 2 √ �� 12t + 3 √ � 2t = √ �� 25(2t) - 2 √ �� 4(3t) + 3 √ � 2t

= √ �� 25 √ � 2t - 2 √ � 4 √ � 3t + 3 √ � 2t = 5 √ � 2t - 4 √ � 3t + 3 √ � 2t = 8 √ � 2t - 4 √ � 3t


14. P = √ �� 50 + √ � 8 + √ �� 18 + √ � 8 = √ �� 25(2) + 2 √ �� 4(2) + √ �� 9(2)

= √ �� 25 √ � 2 + 2 √ � 4 √ � 2 + √ � 9 √ � 2 = 5 √ � 2 + 4 √ � 2 + 3 √ � 2 = 12 √ � 2 in.


15. 4 √ � 3 + 2 √ � 3 = 6 √ � 3

16. 1 __ 2 √ �� 72 - 12 = 1 __

2 √ �� 36(2) - 12

= 1 __ 2 √ �� 36 √ � 2 - 12

= 3 √ � 2 -12

17. 2 √ � 11 + √ � 11 - 6 √ � 11 = -3 √ � 11

18. 6 √ � 7 + 7 √ � 6 = 6 √ � 7 + 7 √ � 6

19. -3 √ � n - √ � n = -4 √ � n

20. 2 √ � 2y + 3 √ � 2y - 2 √ � 3y = 5 √ � 2y - 2 √ � 3y

21. √ �� 175 + √ �� 28 = √ �� 25(7) + √ �� 4(7) = √ �� 25 √ � 7 + √ � 4 √ � 7 = 5 √ � 7 + 2 √ � 7 = 7 √ � 7

22. 2 √ �� 80 - √ �� 20 = 2 √ �� 16(5) - √ �� 4(5) = 2 √ �� 16 √ � 5 - √ � 4 √ � 5 = 8 √ � 5 - 2 √ � 5 = 6 √ � 5

23. 5 √ � 8 - √ �� 32 + 2 √ �� 18 = 5 √ �� 4(2) - √ �� 16(2) + 2 √ �� 9(2)

= 5 √ � 4 √ � 2 - √ �� 16 √ � 2 + 2 √ � 9 √ � 2 = 10 √ � 2 - 4 √ � 2 + 6 √ � 2 = 12 √ � 2

24. √ �� 150r + √ �� 54r = √ �� 25(6r) + √ �� 9(6r) = √ �� 25 √ � 6r + √ � 9 √ � 6r = 5 √ � 6r + 3 √ � 6r = 8 √ � 6r

25. √ �� 63x - 4 √ �� 27x = √ �� 9(7x) - 4 √ �� 9(3x) = √ � 9 √ � 7x - 4 √ � 9 √ � 3x = 3 √ � 7x - 12 √ � 3x

26. √ �� 48p + 3 √ �� 18p - 2 √ �� 27p = √ �� 16(3p) + 3 √ �� 9(2p) - 2 √ �� 9(3p)

= √ �� 16 √ �� 3p + 3 √ � 9 √ �� 2p - 2 √ � 9 √ �� 3p

= 4 √ �� 3p + 9 √ �� 2p - 6 √ �� 3p

= 9 √ �� 2p - 2 √ �� 3p

27. √ �� 180j - √ �� 45j = √ �� 36(5j) - √ �� 9(5j) = √ �� 36 √ � 5j - √ � 9 √ � 5j

= 6 √ � 5j - 3 √ � 5j

= 3 √ � 5j

28. 3 √ �� 90c - √ �� 40c = 3 √ �� 9(10c) - √ �� 4(10c) = 3 √ � 9 √ �� 10c - √ � 4 √ �� 10c = 9 √ �� 10c - 2 √ �� 10c = 7 √ �� 10c

29. 2 √ �� 75m - √ �� 12m - √ �� 108m = 2 √ �� 25(3m) - √ �� 4(3m) - √ �� 36(3m)

= 2 √ �� 25 √ �� 3m - √ � 4 √ �� 3m - √ �� 36 √ �� 3m = 10 √ �� 3m - 2 √ �� 3m - 6 √ �� 3m = 2 √ �� 3m

30. P = 1 + √ � 8 + √ � 2 + 1 + √ � 2 = 2 + 2 √ � 2 + √ �� 4(2)

= 2 + 2 √ � 2 + √ � 4 √ � 2 = 2 + 2 √ � 2 + 2 √ � 2 = 2 + 4 √ � 2 mi

31. 5 √ � 7 + 7 √ � 7 = 12 √ � 7

32. 18 √ �� ab - 10 √ �� ab = 8 √ �� ab

33. -3 √ � 3 + 3 √ � 3 = 0

34. √ �� 98 + √ �� 128 = √ �� 49(2) + √ �� 64(2) = √ �� 49 √ � 2 + √ �� 64 √ � 2 = 7 √ � 2 + 8 √ � 2 = 15 √ � 2

35. √ �� 300 - √ �� 27 = √ �� 100(3) - √ �� 9(3) = √ �� 100 √ � 3 - √ � 9 √ � 3 = 10 √ � 3 - 3 √ � 3 = 7 √ � 3

36. √ �� 45x + √ �� 500x = √ �� 9(5x) + √ �� 100(5x) = √ � 9 √ � 5x + √ �� 100 √ � 5x = 3 √ � 5x + 10 √ � 5x = 13 √ � 5x

37. 5 __ 2 √ � 8 +

√ �� 32 ____

2 = 5 __

2 √ �� 4(2) + 1 __

2 √ �� 16(2)

= 5 __ 2 √ � 4 √ � 2 + 1 __

2 √ �� 16 √ � 2

= 5 √ � 2 + 2 √ � 2 = 7 √ � 2

38. 1 __ 6 √ �� 18 -

√ � 2 ___ 2 = 1 __

6 √ �� 9(2) - 1 __

2 √ � 2

= 1 __ 6 √ � 9 √ � 2 - 1 __

2 √ � 2

= 1 __ 2 √ � 2 - 1 __

2 √ � 2

= 0

39a. section A: 3 √ � 11 ; section B: 2 √ � 11 ; section C: 5 √ � 11

b. 10 √ � 11

c. Because the areas found in parts a and b must be equal, the model shows that:

3 √ � 11 + 2 √ � 11 + 5 √ � 11 = (3 + 2 + 5) √ � 11 = 10 √ � 11


40. √ �� 450ab - √ �� 50ab = √ �� 225(2ab) - √ �� 25(2ab) = √ �� 225 √ �� 2ab - √ �� 25 √ �� 2ab = 15 √ �� 2ab - 5 √ �� 2ab = 10 √ �� 2ab

41. √ �� 12 + √ �� 125 + √ �� 25 = √ �� 4(3) + √ �� 25(5) + 5 = √ � 4 √ � 3 + √ �� 25 √ � 5 + 5 = 2 √ � 3 + 5 √ � 5 + 5

42. √ �� 338 - √ �� 18 = √ �� 169(2) - √ �� 9(2) = √ �� 169 √ � 2 - √ � 9 √ � 2 = 13 √ � 2 - 3 √ � 2 = 10 √ � 2

43. √ �� 700x - √ �� 28x - √ �� 70x = √ �� 100(7x) - √ �� 4(7x) - √ �� 70x

= √ �� 100 √ � 7x - √ � 4 √ � 7x - √ �� 70x = 10 √ � 7x - 2 √ � 7x - √ �� 70x = 8 √ � 7x - √ �� 70x

44. -3 √ �� 90 - 3 √ �� 160 = -3 √ �� 9(10) - 3 √ �� 16(10) = -3 √ � 9 √ �� 10 - 3 √ �� 16 √ �� 10 = -9 √ �� 10 - 12 √ �� 10 = -21 √ �� 10

45. 7 √ �� 80k + 2 √ �� 20k + √ �� 45k = 7 √ �� 16(5k) + 2 √ �� 4(5k) + √ �� 9(5k)

= 7 √ �� 16 √ � 5k + 2 √ � 4 √ � 5k + √ � 9 √ � 5k = 28 √ � 5k + 4 √ � 5k + 3 √ � 5k = 35 √ � 5k

46. √ �� 24abc + √ �� 600abc = √ �� 4(6abc) + √ �� 100(6abc) = √ � 4 √ �� 6abc + √ �� 100 √ �� 6abc = 12 √ �� 6abc

47. √ �� 12 + √ �� 20 + √ �� 27 + √ �� 45 = √ �� 4(3) + √ �� 4(5) + √ �� 9(3) + √ �� 9(5)

= √ � 4 √ � 3 + √ � 4 √ � 5 + √ � 9 √ � 3 + √ � 9 √ � 5 = 2 √ � 3 + 2 √ � 5 + 3 √ � 3 + 3 √ � 5 = 5 √ � 3 + 5 √ � 5

48. A and C are incorrect. In A, the radicands were added. In C, the radicals were not like radicals but they were incorrectly combined by subtracting the radicands.

49. Possible answer: Like radicals have the same number, variable, or numbers and variables under the radical sign; examples: √ � 2 and 3 √ � 2 ; nonexamples: √ � 5 and √ � 3 .

50. 5 √ �� ab + 2 √ � x - 3 √ � a = 7 √ �� ab - 3 √ � a 2 √ � x = 7 √ �� ab - 5 √ �� ab 2 √ � x = 2 √ �� ab x = ab

51. 4 √ � x - √ � yx = √ � x - √ � y √ � x = √ � x - 4 √ � x √ � y √ � x = 3 √ � x √ � y = 3 y = 9

52. 5 √ � 2 - √ � x + √ � 2 = 4 √ � 2 √ � x = 5 √ � 2 + √ � 2 - 4 √ � 2 √ � x = 2 √ � 2 √ � x = √ � 4 √ � 2 √ � x = √ � 8 x = 8

53. √ � x + 8 √ � 2 = 11 √ � 2 √ � x = 3 √ � 2 √ � x = √ � 9 √ � 2 √ � x = √ �� 18 x = 18

54. 3 √ � 3 + 2 √ � 3 + √ � x = 9 √ � 3 √ � x = 4 √ � 3 √ � x = √ �� 16 √ � 3 √ � x = √ �� 48 x = 48

55. 2x - √ � y = -4x - √ � y = -6x

√ � y = √ �� 36 x 2 y = 36 x 2

56a. d = 2r250 = 2rr = 125 ft

b. Pythagorean Theorem

57. A = s 2 P = 4s P = 4s = 4( √ �� 48 ) = 4( √ �� 12 ) = 4( √ �� 16(3) ) = 4( √ �� 4(3) )

= 4( √ �� 16 √ � 3 ) = 4 √ � 4 √ � 3 = 4(4 √ � 3 ) = 8 √ � 3 in. = 16 √ � 3 in.16 √ � 3 + 8 √ � 3 = 24 √ � 3 in.

58. The radical is similar to a variable. To add or subtract, combine coefficients.

59. B; the radicands have no common factors and are hence not like radicals.

60. F; -5 √ � 7x + 6 √ � 7x = √ � 7x

61. A; √ �� 18 - √ � 2 = √ �� 9(2) - √ � 2 = √ � 9 √ � 2 - √ � 2 = 3 √ � 2 - √ � 2 = 2 √ � 2


62. 5 √ �� x - 5 + 2 √ �� x - 5 = 7 √ �� x - 5

63. x √ � x + 2 √ � 3 = √ � x (x + 2)

64. 4 √ �� x - 3 + √ �� 25x - 75 = 4 √ �� x - 3 + √ �� 25(x - 3) = 4 √ �� x - 3 + √ �� 25 √ �� x - 3 = 4 √ �� x - 3 + 5 √ �� x - 3 = 9 √ �� x - 3

65. 2 √ �� x + 7 - √ �� 4x + 28 = 2 √ �� x + 7 - √ �� 4(x + 7) = 2 √ �� x + 7 - √ � 4 √ �� x + 7 = 2 √ �� x + 7 - 2 √ �� x + 7 = 0


66. √ �� 4 x 3 + 24 x 2 + √ �� x 3 + 6 x 2

= √ �� 4 x 2 (x + 6) + √ �� x 2 (x + 6)

= √ �� 4 x 2 √ �� x + 6 + √ � x 2 √ �� x + 6 = 2x √ �� x + 6 + x √ �� x + 6 = 3x √ �� x + 6

67. √ �� x 3 - x 2 + √ �� 4x - 4 = √ �� x 2 (x - 1) + √ �� 4(x - 1)

= √ � x 2 √ �� x - 1 + √ � 4 √ �� x - 1 = x √ �� x - 1 + 2 √ �� x - 1 = (x + 2) √ �� x - 1

68. √ �� x 3 + 2 x 2 - √ �� x + 2

= √ �� x 2 (x + 2) - √ �� x + 2

= √ � x 2 √ �� x + 2 - √ �� x + 2 = x √ �� x + 2 - √ �� x + 2 = (x - 1) √ �� x + 2

69. √ �� 9x + 9 - √ �� x 3 + 2 x 2 = √ �� 9(x + 1) - √ �� x 2 (x + 2)

= √ � 9 √ �� x + 1 - √ � x 2 √ �� x + 2 = 3 √ �� x + 1 - x √ �� x + 2

70. A = 1 __ 2 h( b 1 + b 2 )

= 1 __ 2 (4 √ � 5 )( √ �� 20 + √ �� 45 )

= 1 __ 2 (4 √ � 5 )( √ �� 4(5) + √ �� 9(5) )

= 1 __ 2 (4 √ � 5 )( √ � 4 √ � 5 + √ � 9 √ � 5 )

= 1 __ 2 (4 √ � 5 )(2 √ � 5 + 3 √ � 5 )

= 1 __ 2 (4 √ � 5 )(5 √ � 5 )

= 1 __ 2 (20)(5)

= 50 cm 2


71. m AB = 4 - 1 ______ 4 - 1

= 1, m BC = 3 - 4 _______ -2 - 4 = 1 __

6

m CD = 0 - 3 _________ -5 - (-2)

= 1, m AD = 0 - 1 _______ -5 - 1 = 1 __

6

Since m AB = m CD , −− AB ‖

−− CD .Since m BC = m AD ,

−− BC ‖ −− AD .

Since both pairs of opposite sides are parallel, ABCD is a parallelogram.

72. m XZ = 4 - 0 ________ 2 - (-1)

= 4 __ 3 ;

m YZ = 0 - (-3)

________ -1 - 3 = - 3 __

4 ;

Since the product of the slopes is -1, −− XZ ⊥

−−− YZ, XYZ is a right triangle.

73. P(roll 6 and toss heads) = 1 _ 6 · 1 _

2

= 1 _ 12

74. y = √ �� 4x - 2 4x - 2 ≥ 0 4x ≥ 2

x ≥ 1 __ 2

Domain: { x | x ≥ 1 __ 2 }

75. y = -2 √ �� x + 3 x + 3 ≥ 0 x ≥ -3Domain: { x | x ≥ -3 }

76. y = 1 + √ �� x + 6 x + 6 ≥ 0 x ≥ -6Domain: { x | x ≥ -6 }

11-8 MULTIPLYING AND DIVIDING RADICAL EXPRESSIONS, PAGES 816–821


1a. √ � 5 √ �� 10 = √ �� 5(10) = √ �� 50 = √ �� 25(2)

= √ �� 25 √ � 2 = 5 √ � 2

b. (3 √ � 7 ) 2 = (3 √ � 7 ) (3 √ � 7 )

= 9 √ � 7 √ � 7 = 9 √ �� 7(7) = 9(7) = 63

1c. √ �� 2m √ �� 14m = √ �� 2m(14m)

= √ �� 28 m 2

= √ �� 4 m 2 (7)

= √ �� 4 m 2 √ � 7 = 2m √ � 7

2a. √ � 6 ( √ � 8 - 3) = √ � 6 √ � 8 - 3 √ � 6 = √ �� 6(8) - 3 √ � 6

= √ �� 48 - 3 √ � 6 = √ �� (16)3 - 3 √ � 6

= √ �� 16 √ � 3 - 3 √ � 6 = 4 √ � 3 - 3 √ � 6

b. √ � 5 ( √ �� 10 + 4 √ � 3 ) = √ � 5 √ �� 10 + 4 √ � 5 √ � 3 = √ �� 5(10) + 4 √ �� 5(3)

= √ �� 50 + 4 √ �� 15 = √ �� 25(2) + 4 √ �� 15

= √ �� 25 √ � 2 + 4 √ �� 15 = 5 √ � 2 + 4 √ �� 15

c. √ � 7k ( √ � 7 - 5) = √ � 7k √ � 7 - 5 √ � 7k = √ �� 7(7k) - 5 √ � 7k

= √ �� 49k - 5 √ � 7k = √ �� 49 √ � k - 5 √ � 7k = 7 √ � k - 5 √ � 7k


d. 5 √ � 5 (-4 + 6 √ � 5 ) = -20 √ � 5 + 30 √ � 5 √ � 5 = -20 √ � 5 + 30 √ �� 5(5)

= -20 √ � 5 + 30 √ �� 25 = -20 √ � 5 + 30(5) = 150 - 20 √ � 5

3a. (3 + √ � 3 ) (8 - √ � 3 ) = 24 - 3 √ � 3 + 8 √ � 3 - √ � 3 √ � 3 = 24 + 5 √ � 3 - √ �� 3(3)

= 24 + 5 √ � 3 - √ � 9 = 24 + 5 √ � 3 - 3 = 21 + 5 √ � 3

b. (9 + √ � 2 ) 2 = (9 + √ � 2 ) (9 + √ � 2 )

= 81 + 9 √ � 2 + 9 √ � 2 + √ � 2 √ � 2 = 81 + 18 √ � 2 + √ �� 2(2)

= 81 + 18 √ � 2 + √ � 4 = 81 + 18 √ � 2 + 2 = 83 + 18 √ � 2

c. (3 - √ � 2 ) 2 = (3 - √ � 2 ) (3 - √ � 2 )

= 9 - 3 √ � 2 - 3 √ � 2 + √ � 2 √ � 2 = 9 - 6 √ � 2 + √ �� 2(2)

= 9 - 6 √ � 2 + √ � 4 = 9 - 6 √ � 2 + 2 = 11 - 6 √ � 2

d. (4 - √ � 3 ) ( √ � 3 + 5) = 4 √ � 3 + 20 - √ � 3 √ � 3 - 5 √ � 3 = 20 - √ �� 3(3) - √ � 3

= 20 - √ � 9 - √ � 3 = 20 - 3 - √ � 3 = 17 - √ � 3

4a. √ �� 13

____ √ � 5

= √ �� 13

____ √ � 5

( √ � 5

___ √ � 5

)

= √ �� 65

____ √ �� 25

= √ �� 65

____ 5

b. √ �� 7a

____ √ �� 12

= √ �� 7a

_____ √ �� 4(3)

= √ �� 7a

____ 2 √ � 3

= √ �� 7a

____ 2 √ � 3

( √ � 3

___ √ � 3

)

= √ �� 21a

_____ 2 √ � 9

= √ �� 21a

_____ 6

c. 2 √ �� 80 _____

√ � 7 =

2 √ �� 16(5) _______

√ � 7

= 8 √ � 5

____ √ � 7

( √ � 7 ___ √ � 7

)

= 8 √ �� 35

_____ √ �� 49

= 8 √ �� 35

_____ 7


1. √ � 5 ___ √ � 5

is equal to 1, so multiplying by √ � 5 ___ √ � 5

does not

change the value of the original expression.

2. Possible answer:


1. √ � 2 √ � 3 = √ �� 2(3) = √ � 6

2. √ � 3 √ � 8 = √ �� 3(8) = √ �� 24 = √ �� 4(6)

= √ � 4 √ � 6 = 2 √ � 6

3. (5 √ � 5 ) 2 = (5 √ � 5 ) (5 √ � 5 )

= 25 √ �� 5(5)

= 25 √ �� 25 = 125

4. (4 √ � 2 ) 2 = (4 √ � 2 ) (4 √ � 2 )

= 16 √ �� 2(2)

= 16 √ � 4 = 32

5. 3 √ �� 3a √ �� 10 = 3 √ �� 3a(10)

= 3 √ �� 30a

6. 2 √ �� 15p √ �� 3p = 2 √ �� 45 p 2

= 2 √ �� 9 p 2 (5)

= 2 √ �� 9 p 2 √ � 5

= 6p √ � 5

7. √ � 6 (2 + √ � 7 ) = 2 √ � 6 + √ � 6 √ � 7 = 2 √ � 6 + √ �� 6(7)

= 2 √ � 6 + √ �� 42

8. √ � 3 (5 - √ � 3 ) = 5 √ � 3 - √ � 3 √ � 3 = 5 √ � 3 - √ �� 3(3)

= 5 √ � 3 - √ � 9 = 5 √ � 3 - 3

9. √ � 7 ( √ � 5 - √ � 3 ) = √ � 7 √ � 5 - √ � 7 √ � 3 = √ �� 7(5) - √ �� 7(3)

= √ �� 35 - √ �� 21

10. √ � 2 ( √ �� 10 + 8 √ � 2 ) = √ � 2 √ �� 10 + 8 √ � 2 √ � 2 = √ �� 2(10) + 8 √ �� 2(2)

= √ �� 20 + 8 √ � 4 = √ �� 4(5) + 8(2)

= √ � 4 √ � 5 + 16 = 2 √ � 5 + 16

11. √ � 5y ( √ �� 15 + 4) = √ �� 5y(15) + 4 √ � 5y = √ �� 75y + 4 √ � 5y

= √ �� 25(3y) + 4 √ � 5y

= √ �� 25 √ � 3y + 4 √ � 5y

= 5 √ � 3y + 4 √ � 5y


12. √ � 2t ( √ � 6t - √ � 2t ) = √ �� 2t(6t) - √ �� 2t(2t)

= √ �� 12 t 2 - 2t

= √ �� 4 t 2 (3) - 2t

= √ �� 4 t 2 √ � 3 - 2t = 2t √ � 3 - 2t

13. (2 + √ � 2 ) (5 + √ � 2 ) = 10 + 2 √ � 2 + 5 √ � 2 + 2 = 12 + 7 √ � 2

14. (4 + √ � 6 ) (3 - √ � 6 ) = 12 - 4 √ � 6 + 3 √ � 6 - 6 = 6 - √ � 6

15. ( √ � 3 - 4) ( √ � 3 + 2) = 3 + 2 √ � 3 - 4 √ � 3 - 8 = -5 - 2 √ � 3

16. (5 + √ � 3 ) 2 = (5 + √ � 3 ) (5 + √ � 3 )

= 25 + 5 √ � 3 + 5 √ � 3 + 3 = 28 + 10 √ � 3

17. ( √ � 6 - 5 √ � 3 ) 2 = ( √ � 6 - 5 √ � 3 ) ( √ � 6 - 5 √ � 3 )

= 6 - 5 √ �� 3(6) - 5 √ �� 3(6) + 25(3)

= 6 - 5 √ �� 18 - 5 √ �� 18 + 75 = 81 - 10 √ �� 18 = 81 - 10 √ �� 9(2)

= 81 - 30 √ � 2

18. (6 + 3 √ � 2 ) 2 = (6 + 3 √ � 2 ) (6 + 3 √ � 2 )

= 36 + 18 √ � 2 + 18 √ � 2 + 9(2) = 36 + 36 √ � 2 + 18 = 54 + 36 √ � 2

19. √ �� 13

____ √ � 2

= √ �� 13

____ √ � 2

( √ � 2 ___ √ � 2

)

= √ �� 26

____ 2

20. √ �� 20

____ √ � 8

= √ �� 4(5)

_____ √ �� 4(2)

= 2 √ � 5

____ 2 √ � 2

= √ � 5

___ √ � 2

= √ � 5

___ √ � 2

( √ � 2 ___ √ � 2

)

= √ �� 10

____ 2

21. √ � 11 ____ 6 √ � 3

= √ � 11 ____ 6 √ � 3

( √ � 3

___ √ � 3

)

= √ �� 33

____ 6(3)

= √ �� 33

____ 18

22. √ �� 28

____ √ � 3s

= √ �� 4(7)

_____ √ � 3s

= 2 √ � 7 ____ √ � 3s

= 2 √ � 7 ____ √ � 3s

( √ � 3s

____ √ � 3s

)

= 2 √ �� 21s

______ 3s

23. 2 ___ √ � 7

= 2 ___ √ � 7

( √ � 7 ___ √ � 7

)

= 2 √ � 7 ____ 7

24. 3 ___ √ � 6

= 3 ___ √ � 6

( √ � 6

___ √ � 6

)

= √ � 6

___ 2

25. 1/ √ � 5x = 1 ____ √ � 5x

( √ � 5x

____ √ � 5x

)

= √ � 5x

____ 5x

26. √ � 3

___ √ � x

= √ � 3

___ √ � x

( √ � x ___ √ � x

)

= √ � 3x

____ x


27. √ � 3 √ � 5 √ � 6 = √ �� 3(5)(6) = √ �� 90 = √ �� 9(10)

= 3 √ �� 10

28. (3 √ � 6 ) (5 √ � 6 ) = 15 √ �� 6(6) = 15(6) = 90

29. (2 √ � 2 ) 2 = (2 √ � 2 ) (2 √ � 2 )

= 4 √ �� 2(2) = 4(2) = 8

30. (3 √ � 6 ) 2 = (3 √ � 6 ) (3 √ � 6 )

= 9 √ �� 6(6) = 9(6) = 54

31. √ �� 21d (2 √ �� 3d ) = 2 √ �� 3(21) d 2

= 2 √ �� 63 d 2

= 2 √ �� 9 d 2 (7)

= 2 √ �� 9 d 2 √ � 7 = 6d √ � 7

32. 4 √ �� 5n (2 √ �� 5n ) (3 √ �� 3n ) = 24 √ �� (5n) 2 √ �� 3n = 24(5n) √ �� 3n = 120n √ �� 3n

33. √ � 5 (4 - √ �� 10 ) = 4 √ � 5 - √ �� 5(10) = 4 √ � 5 - √ �� 50 = 4 √ � 5 - √ �� 25(2)

= 4 √ � 5 - 5 √ � 2

34. √ � 2 ( √ � 6 + 2) = √ �� 2(6) + 2 √ � 2 = √ �� 12 + 2 √ � 2 = √ �� 4(3) + 2 √ � 2

= 2 √ � 3 + 2 √ � 2

35. √ � 2 ( √ � 6 - √ �� 10 ) = √ �� 2(6) - √ �� 2(10) = √ �� 12 - √ �� 20 = √ �� 4(3) - √ �� 4(5)

= 2 √ � 3 - 2 √ � 5

36. 3 √ � 3 ( √ � 8 - 2 √ � 6 ) = 3 √ �� 3(8) - 6 √ �� 3(6) = 3 √ �� 24 - 6 √ �� 18 = 3 √ �� 4(6) - 6 √ �� 9(2)

= 6 √ � 6 - 18 √ � 2

37. √ � 3f ( √ � 3 + 12) = √ �� 3f(3) + 12 √ � 3f = √ � 9f + 12 √ � 3f = 3 √ � f + 12 √ � 3f

38. √ �� 8m ( √ �� 10 + √ �� 2m ) = √ �� 8m(10) + √ �� 8m(2m)

= √ �� 80m + √ �� 16 m 2 = √ �� 16(5m) + 4m

= 4 √ �� 5m + 4m

39. (15 + √ �� 15 ) (4 + √ �� 15 )

= 60 + 15 √ �� 15 + 4 √ �� 15 + 15= 75 + 19 √ �� 15


40. ( √ � 6 + 4) ( √ � 2 - 7) = √ �� 6(2) - 7 √ � 6 + 4 √ � 2 - 28 = √ �� 12 - 7 √ � 6 + 4 √ � 2 - 28 = √ �� 4(3) - 7 √ � 6 + 4 √ � 2 - 28

= 2 √ � 3 - 7 √ � 6 + 4 √ � 2 - 28

41. (3 - √ � 2 ) (4 + √ � 2 ) = 12 + 3 √ � 2 - 4 √ � 2 - √ � 2 = 10 - √ � 2

42. ( √ � 5 - 5) 2 = ( √ � 5 - 5) ( √ � 5 - 5)

= 5 - 5 √ � 5 - 5 √ � 5 + 25 = 30 - 10 √ � 5

43. ( √ � 3 + 8) 2 = ( √ � 3 + 8) ( √ � 3 + 8)

= 3 + 8 √ � 3 + 8 √ � 3 + 64 = 67 + 16 √ � 3

44. (2 √ � 3 + 4 √ � 5 ) 2 = (2 √ � 3 + 4 √ � 5 ) (2 √ � 3 + 4 √ � 5 )

= 4(3) + 8 √ �� 3(5) + 8 √ �� 3(5) + 16(5)

= 92 + 16 √ �� 15

45. √ �� 75

____ √ � 2

= √ �� 75

____ √ � 2

( √ � 2 ___ √ � 2

)

= √ �� 150

_____ 2

= √ �� 25(6)

______ 2

= 5 √ � 6

____ 2

46. √ � 5

____ 4 √ � 8

= √ � 5 ______

4 √ �� 4(2)

= √ � 5

____ 8 √ � 2

= √ � 5

____ 8 √ � 2

( √ � 2 ___ √ � 2

)

= √ �� 5(2)

_____ 8(2)

= √ �� 10

____ 16

47. √ �� 27 ____ 3 √ � x

= √ �� 27 ____ 3 √ � x

( √ � x ___ √ � x

)

= √ �� 27x ______ 3x

= √ �� 9(3x)

______ 3x

= 3 √ � 3x

_____ 3x

= √ � 3x

____ x

48. √ �� 48k

_____ √ � 5

= √ �� 48k

_____ √ � 5

( √ � 5

___ √ � 5

)

= √ �� 48(5)k

_______ 5

= √ �� 16(15k)

________ 5

= 4 √ �� 15k

______ 5

49. √ �� 49x

_____ √ � 2

= 7 √ � x ____ √ � 2

= 7 √ � x ____ √ � 2

( √ � 2 ___ √ � 2

)

= 7 √ � 2x _____ 2

50. 3 √ �� 27 _____ √ � b

= 3 √ �� 27 _____ √ � b

( √ � b

___ √ � b

)

= 3 √ �� 27b

______ b

= 3 √ �� 9(3b)

_______ b

= 9 √ �� 3b

_____ b

51. √ �� 12y

_____ √ � 3

= √ ��

12y

____ 3

= √ � 4y = 2 √ � y

52. √ �� 12t

_____ √ � 6

= √ �� 12t ___ 6

= √ � 2t

53. A = (6 √ � 5 ) (6 √ � 5 ) = 36(5) = 180 in 2

54. A = 2 √ � 3 ( √ � 6 ) = 2 √ �� 3(6)

= 2 √ �� 18 = 2 √ �� 9(2)

= 6 √ � 2 m 2

55. A = √ � 5 (6 √ � 2 - 2)

= 6 √ �� 2(5) - 2 √ � 5

= (6 √ �� 10 - 2 √ � 5 ) cm 2

56. √ � 3 ( √ � 2 ___ √ � 7

) = √ �� 3(2)

_____ √ � 7

= √ � 6

___ √ � 7

( √ � 7 ___ √ � 7

)

= √ �� 6(7)

_____ 7

= √ �� 42 ____

7

57. 15 √ �� 10 ______

5 √ � 3 =

3 √ �� 10 _____

√ � 3

= 3 √ �� 10

_____ √ � 3

( √ � 3

___ √ � 3

)

= 3 √ �� 10(3)

_______ 3

= √ �� 30

58. 6 + √ �� 18 ________

3 = 2 +

√ �� 18 ____

3

= 2 + √ �� 9(2)

_____ 3

= 2 + 3 √ � 2 ____ 3

= 2 + √ � 2

59. ( √ � 3 - 4)( √ � 3 + 2)

= 3 + 2 √ � 3 - 4 √ � 3 - 8= -5 - 2 √ � 3

60. √ � 2 (6 + √ �� 12 ) = 6 √ � 2 + √ �� 2(12) = 6 √ � 2 + √ �� 24 = 6 √ � 2 + √ �� 4(6)

= 6 √ � 2 + 2 √ � 6

61. √ � 1 + √ �� 25

_________ √ � 2

= 1 + 5 _____ √ � 2

= 6 ___ √ � 2

= 6 ___ √ � 2

( √ � 2 ___ √ � 2

)

= 6 √ � 2 ____ 2

= 3 √ � 2

62. √ �� 15 + √ �� 10

__________ √ � 5

= √ �� 15

____ √ � 5

+ √ �� 10

____ √ � 5

= √ �� 15 ___ 5 + √ �� 10 ___

5

= √ � 3 + √ � 2

63. √ �� 12 ( √ � 3 + 8) 2 = √ �� 4(3) ( √ � 3 + 8) ( √ � 3 + 8)

= 2 √ � 3 (3 + 8 √ � 3 + 8 √ � 3 + 64)

= 2 √ � 3 (67 + 16 √ � 3 ) = 134 √ � 3 + 32(3) = 134 √ � 3 + 96

64. √ � 3 (4 - 2 √ � 5 ) = 4 √ � 3 - 2 √ �� 5(3)

= 4 √ � 3 - 2 √ �� 15


65. ( √ � x - √ � y ) 2 = ( √ � x - √ � y ) ( √ � x - √ � y )

= x - √ � x √ � y - √ � x √ � y + y = x - 2 √ � xy + y

66. ( √ � x - 5) (3 √ � x + 7) = 3(x) + 7 √ � x - 15 √ � x - 35 = 3x - 8 √ � x - 35

67. ( √ � 3 + √ � x ) 2 = ( √ � 3 + √ � x ) ( √ � 3 + √ � x )

= 3 + √ �� 3(x) + √ �� 3(x) + x

= 3 + 2 √ � 3x + x

68. Current = √ � W ____ √ � R

= √ �� 850

_____ √ � 5

= √ �� 850 ____ 5

= √ �� 170 amps ≈ 13.0 amps

69. P = 2π √ ��

___ 32

= 2π √ �� 3 ___ 32

= 2π ( √ � 3

____ √ �� 32

)

= 2π ( √ � 3 ______

√ �� 16(2) )

= 2π ( √ � 3

____ 4 √ � 2

)

= π √ � 3

_____ 2 √ � 2

= π √ � 3

_____ 2 √ � 2

( √ � 2 ___ √ � 2

)

= π √ � 6

_____ 2(2)

= π √ � 6

_____ 4 s ≈ 1.9 s

70. A = 1 __ 2 bh

= 1 __ 2 ( √ � 3 ) (2 √ � 6 )

= √ �� 3(6)

= √ �� 18 = √ �� 9(2)

= 3 √ � 2 yd 2

71. A = 1 __ 2 bh

= 1 __ 2 (7 √ � 11 ) (7 √ � 11 )

= 1 __ 2 (49)(11)

= 269.5 ft 2

72. A = 1 __ 2 bh

= 1 __ 2 (2 - √ � 3 ) (2 √ � 3 - 3)

= 1 __ 2 (4 √ � 3 - 6 - 2(3) + 3 √ � 3 )

= 1 __ 2 (7 √ � 3 - 12)

= ( 7 √ � 3

____ 2 - 6) cm 2

73. Possible answer: 1 ___ √ � 3

; multiply the fraction by √ � 3

___ √ � 3

.

This will rationalize the denominator, since √ � 3 √ � 3 = 3.

74a. t = √ �� d ___ 16

= √ �� 100 ____ 16

= √ �� 100

_____ √ �� 16

= 10 ___ 4 = 2.5 s

b. 5.6s; it takes more than twice as long to go up the tower as it does to come down.

75. B; (3 √ � 5 ) √ �� 15 = 3 √ �� 75 = 3 √ �� 25(3) = 15 √ � 3

76. H; 4 ____ 3 √ � 2

= 4 ____ 3 √ � 2

( √ � 2 ___ √ � 2

)

= 4 √ � 2 ____ 3(2)

= 2 √ � 2 ____ 3

77. D; (5 √ �� 10 ) 2 = 25(10)

= 250


78. 4 ________ √ � 3 - √ � 2

= 4 ________ √ � 3 - √ � 2

( √ � 3 + √ � 2

________ √ � 3 + √ � 2

)

= 4 ( √ � 3 + √ � 2 )

___________________ ( √ � 3 - √ � 2 ) ( √ � 3 + √ � 2 )

= 4 √ � 3 + 4 √ � 2

__________ 3 - 2

= 4 √ � 3 + 4 √ � 2

79. 8 ________ √ � 3 + √ � 5

= 8 ________ √ � 3 + √ � 5

( √ � 3 - √ � 5

________ √ � 3 - √ � 5

)

= 8 ( √ � 3 - √ � 5 )

___________________ ( √ � 3 + √ � 5 ) ( √ � 3 - √ � 5 )

= 8 √ � 3 - 8 √ � 5

__________ 3 - 5

= 8 √ � 3 - 8 √ � 5

__________ -2

= -4 √ � 3 + 4 √ � 5

80. √ � 5 _________

√ �� 10 + √ � 3 =

√ � 5 _________

√ �� 10 + √ � 3 (

√ �� 10 - √ � 3 _________

√ �� 10 - √ � 3 )

= √ � 5 ( √ �� 10 - √ � 3 )

_____________________ ( √ �� 10 + √ � 3 ) ( √ �� 10 - √ � 3 )

= √ �� 5(10) - √ �� 5(3)

______________ 10 - 3

= √ �� 25(2) - √ �� 15

_____________ 7

= 5 √ � 2 - √ �� 15

__________ 7


81. √ � 2 + √ � 3

________ √ � 2 - √ � 3

= √ � 2 + √ � 3

________ √ � 2 - √ � 3

( √ � 2 + √ � 3

________ √ � 2 + √ � 3

)

= ( √ � 2 + √ � 3 ) ( √ � 2 + √ � 3 )

___________________ ( √ � 2 - √ � 3 ) ( √ � 2 + √ � 3 )

= 2 + √ � 2 √ � 3 + √ � 2 √ � 3 + 3

____________________ 2 - 3

= 5 + 2 √ � 6

________ -1

= -5 - 2 √ � 6

82. √ � 3 ________

√ � 2 + √ � 3 =

√ � 3 ________

√ � 2 + √ � 3 (

√ � 2 - √ � 3 ________

√ � 2 - √ � 3 )

= √ � 3 ( √ � 2 - √ � 3 )

___________________ ( √ � 2 + √ � 3 ) ( √ � 2 - √ � 3 )

= √ �� 3(2) - 3

_________ 2 - 3

= √ � 6 - 3

_______ -1

= 3 - √ � 6

83. √ � 2 ________ √ � 8 + √ � 6

= √ � 2 ________ √ � 8 + √ � 6

( √ � 8 - √ � 6

________ √ � 8 - √ � 6

)

= √ � 2 ( √ � 8 - √ � 6 )

___________________ ( √ � 8 + √ � 6 ) ( √ � 8 - √ � 6 )

= √ �� 2(8) - √ �� 2(6)

_____________ 8 - 6

= √ �� 16 - √ �� 12

__________ 2

= 4 - √ �� 4(3)

_________ 2

= 2 - √ � 3

84. 6 ________ √ � 2 + √ � 3

= 6 ________ √ � 2 + √ � 3

( √ � 2 - √ � 3

________ √ � 2 - √ � 3

)

= 6 ( √ � 2 - √ � 3 )

___________________ ( √ � 2 + √ � 3 ) ( √ � 2 - √ � 3 )

= 6 √ � 2 - 6 √ � 3

__________ 2 - 3

= 6 √ � 3 - 6 √ � 2

85. 2 ________ √ � 6 - √ � 5

= 2 ________ √ � 6 - √ � 5

( √ � 6 + √ � 5

________ √ � 6 + √ � 5

)

= 2 ( √ � 6 + √ � 5 )

___________________ ( √ � 6 - √ � 5 ) ( √ � 6 + √ � 5 )

= 2 √ � 6 + 2 √ � 5

__________ 6 - 5

= 2 √ � 6 + 2 √ � 5

86. A 1 = �w A 2 = �w

= 4 √ � 6 ( √ � 2 ) = 8 √ � 2 (2 √ � 6 ) = 4 √ �� 12 = 16 √ �� 2(6)

= 4 √ �� 4(3) = 16 √ �� 4(3)

= 8 √ � 3 ft 2 = 32 √ � 3 ft 2 A 2 - A 1 = 32 √ � 3 - 8 √ � 3 = 24 √ � 3 ft 2


87. translation of 4 units down

88. rotation about (0, 0) (or vertical stretch, steeper)

89. x 2 + 7x - 30 = x 2 - 3x + 10x - 30

= ( x 2 - 3x) + (10x - 30) = x (x - 3) + 10 (x - 3) = (x + 10) (x - 3)

90. 6 x 2 + 11x + 3 = 6 x 2 + 2x + 9x + 3

= (6 x 2 + 2x) + (9x + 3) = 2x (3x + 1) + 3 (3x + 1) = (3x + 1) (2x + 3)

91. x 2 - 16 = x 2 - 4x + 4x - 16

= ( x 2 - 4x) + (4x - 16) = x (x - 4) + 4 (x - 4) = (x - 4) (x + 4)

92. 3 x 2 + 30x + 75 = 3 ( x 2 + 10x + 25)

= 3 ( ( x 2 + 5x) + (5x + 25) )

= 3 (x (x + 5) + 5 (x + 5) ) = 3 (x + 5) (x + 5) = 3 (x + 5) 2

93. 2 x 4 - 18 = 2 ( x 4 - 9)

= 2 ( x 4 - 3 x 2 + 3 x 2 - 9)

= 2 ( ( x 4 - 3 x 2 ) + (3 x 2 - 9) )

= 2 ( x 2 ( x 2 - 3) + 3 ( x 2 - 3) )

= 2 ( x 2 - 3) ( x 2 + 3)

94. 8 x 3 - 20 x 2 - 12x = 4x (2 x 2 - 5x - 3)

= 4x (2 x 2 - 6x + x - 3)

= 4x ( (2 x 2 - 6x) + (x - 3) )

= 4x (2x (x - 3) + (x - 3) ) = 4x (2x + 1) (x - 3)

95. √ �� 360 = √ �� 36(10) = √ �� 36 √ �� 10 = 6 √ �� 10

96. √ �� 72 ___ 16

= √ �� 72 ____ √ �� 16

= √ �� 36(2)

______ √ �� 16

= √ �� 36 √ � 2

_______ √ �� 16

= 6 √ � 2 ____ 4

= 3 √ � 2 ____ 2

97. √ ��

49 x 2 ____ 64 y 4

= √ �� 49 x 2

______ √ �� 64 y 4

= 7x ___ 8 y 2


98. √ ��

50 a 7 ____ 9 a 3

= √ ��

50 a 4 ____ 9

= √ �� 50 a 4

______ √ � 9

= √ �� 25 a 4 (2)

________ √ � 9

= 5 a 2 √ � 2 ______ 3

11-9 SOLVING RADICAL EQUATIONS, PAGES 822–829


1a. √ � x = 6 ( √ � x ) 2 = (6) 2 x = 36

b. 9 = √ �� 27x

(9) 2 = ( √ �� 27x ) 2

81 = 27x 3 = x

c. √ � 3x = 1

( √ � 3x ) 2 = (1) 2

3x = 1

x = 1 __ 3

2a. √ � x - 2 = 1 √ � x = 3 ( √ � x ) 2 = (3) 2 x = 9

b. √ �� x + 7 = 5

( √ �� x + 7 ) 2 = (5) 2

x + 7 = 25 x = 18

c. √ �� 3x + 7 - 1 = 3 √ �� 3x + 7 = 4

( √ �� 3x + 7 ) 2 = (4) 2

3x + 7 = 16 3x = 9 x = 3

3a. 2 √ � x = 22 √ � x = 11 ( √ � x ) 2 = 11 2 x = 121

b. 2 = √ � x ___ 4

8 = √ � x (8) 2 = ( √ � x ) 2 x = 64

c. 2 √ � x ____ 5 = 4

√ � x = 10 ( √ � x ) 2 = (10) 2 x = 100

4a. √ �� 3x + 2 = √ �� x + 6

( √ �� 3x + 2 ) 2 = ( √ �� x + 6 )

2

3x + 2 = x + 6 2x = 4 x = 2

b. √ �� 2x - 5 - √ � 6 = 0 √ �� 2x - 5 = √ � 6

( √ �� 2x - 5 ) 2 = ( √ � 6 )

2

2x - 5 = 6 2x = 11

x = 11 ___ 2

5a. 11 + √ � 5x = 6 √ � 5x = -5

( √ � 5x ) 2 = (-5) 2

5x = 25 x = 5Check: ____________ 11 + √ � 5x = 6 11 + √ �� 5(5) 6 11 + 5 6 16 6 ✗; Hence, no solution.

b. x = √ �� -3x - 2

(x) 2 = ( √ �� -3x - 2 ) 2

x 2 = -3x - 2 x 2 + 3x + 2 = 0(x+ 2)(x + 1) = 0x + 2 = 0 or x + 1 = 0 x = -2 or x = -1Check: ____________ x = √ �� -3x - 2 -2 √ �� -3(-2) - 2

-2 √ �� 6 - 2 -2 √ � 4 -2 2 ✗

____________ x = √ �� -3x - 2 -1 √ �� -3(-1) - 2

-1 √ �� 3 -2 -1 √ � 1 -1 1 ✗; So, no solution.

c. x - 2 = √ � x (x - 2) 2 = ( √ � x ) 2 x 2 - 4x + 4 = x x 2 - 5x + 4 =0(x - 1)(x - 4) = 0x - 1 = 0 or x - 4 = 0 x = 1 or x = 4Check: __________ x - 2 = √ � x 1 - 2 √ � 1 -1 1 ✗ x - 2 = √ � x

4 - 2 √ � 4 2 2 ✓; The only solution is 4.


6. A = �w

15 = ( √ �� x + 1 ) (5)

3 = √ �� x + 1

(3) 2 = ( √ �� x + 1 ) 2

9 = x + 1 8 = x� = √ �� x + 1 = 3 cm


1. Possible answer: Method 1 is preferable because 21 is easily divided by 3 and dividing by 3 first keeps the numbers small.

2. Subtract 3 from both sides. After doing this, square both sides to eliminate the radical.

3. Possible answer:


1. No; it does not contain a variable under the radical sign.

2. √ � x = 7 ( √ � x ) 2 = (7) 2 x = 49

3. 4 = √ �� -2y

(4) 2 = ( √ �� -2y ) 2

16 = -2y -8 = y

4. √ �� 20a = 10

( √ �� 20a ) 2 = (10) 2

20a = 100 a = 5

5. 12 = √ �� -x (12) 2 = ( √ �� -x ) 2 144 = -x-144 = x

6. √ � x + 6 = 11 √ � x = 5 ( √ � x ) 2 = (5) 2 x = 25

7. √ �� 2x - 5 = 7

( √ �� 2x - 5 ) 2 = (7) 2

2x - 5 = 49 2x = 54 x = 27

8. √ �� 2 - a = 3

( √ �� 2 - a ) 2 = (3) 2

2 - a = 9 -a = 7 a = -7

9. √ � 2x - 3 = 7 √ � 2x = 10

( √ � 2x ) 2 = (10) 2

2x = 100 x = 50

10. √ �� x - 2 = 3

( √ �� x - 2 ) 2 = (3) 2

x - 2 = 9 x = 11

11. √ �� x + 3 = 1

( √ �� x + 3 ) 2 = (1) 2

x + 3 = 1 x = -2

12. √ �� x - 1 = 2

( √ �� x - 1 ) 2 = (2) 2

x - 1 = 4 x = 5

13. √ �� 4y + 13 - 1 = 6 √ �� 4y + 13 = 7

( √ �� 4y + 13 ) 2 = (7) 2

4y + 13 = 49 4y = 36 y = 9

14. -2 √ � x = -10 √ � x = 5 ( √ � x ) 2 = (5) 2 x = 25

15. √ � a

___ 2 = 4

√ � a = 8 ( √ � a ) 2 = (8) 2 a = 64

16. 5 √ �� -x = 20 √ �� -x = 4 ( √ �� -x ) 2 = (4) 2 -x = 16 x = -16

17. 3 √ � x ____ 4 = 3

√ � x = 4 ( √ � x ) 2 = (4) 2 x = 16

18. 5 √ � x ____ 6 = 10

√ � x = 12 ( √ � x ) 2 = (12) 2 x = 144

19. 2 √ � x = 8 √ � x = 4 ( √ � x ) 2 = (4) 2 x = 16

20. √ � x ___ 3 = 3

√ � x = 9 ( √ � x ) 2 = (9) 2 x = 81

21. 3 √ � x ____ 2 = 1

√ � x = 2 __ 3

( √ � x ) 2 = ( 2 __ 3 )

2

x = 4 __ 9

22. 13 √ � 2x = 26 √ � 2x = 2

( √ � 2x ) 2 = (2) 2

2x = 4 x = 2

23. √ � x ___ 5 = 2

√ � x = 10 ( √ � x ) 2 = (10) 2 x = 100

24. √ �� x - 7 _______

3 = 1

√ �� x - 7 = 3

( √ �� x - 7 ) 2 = 3 2

x - 7 = 9 x = 16

25. 4 √ �� 2x - 1 = 12 √ �� 2x - 1 = 3

( √ �� 2x - 1 ) 2 = (3) 2

2x - 1 = 9 x = 5

26. √ �� 5 - x = √ �� 6x - 2

( √ �� 5 - x ) 2 = ( √ �� 6x - 2 )

2

5 - x = 6x - 2 7 = 7x 1 = x

27. √ �� x + 7 = √ �� 3x - 19

( √ �� x + 7 ) 2 = ( √ �� 3x - 19 )

2

x + 7 = 3x - 19 26 = 2x 13 = x


28. 0 = √ � 2x - √ �� x + 3 √ �� x + 3 = √ � 2x

( √ �� x + 3 ) 2 = ( √ � 2x )

2

x + 3 = 2x 3 = x

29. √ �� x - 5 = √ �� 7 - x

( √ �� x - 5 ) 2 = ( √ �� 7 - x )

2

x - 5 = 7 - x 2x = 12 x = 6

30. √ �� -x = √ �� 2x + 1

( √ �� -x ) 2 = ( √ �� 2x + 1 ) 2

-x = 2x + 1 -3x = 1

x = - 1 __ 3

31. √ �� 3x + 1 - √ �� 2x + 3 = 0 √ �� 3x + 1 = √ �� 2x + 3

( √ �� 3x + 1 ) 2 = ( √ �� 2x + 3 )

2

3x + 1 = 2x + 3 x = 2

32. √ �� x - 5 + 5 = 0 √ �� x - 5 = -5

( √ �� x - 5 ) 2 = (5) 2

x - 5 = 25 x = 30Check: ______________ √ �� x - 5 + 5 = 0 √ �� 30 - 5 + 5 0 √ �� 25 + 5 0 5 + 5 0 10 0✗ no solution

33. √ � 3x + 5 = 3 √ � 3x = -2

( √ � 3x ) 2 = (-2) 2

3x = 4

x = 4 __ 3

Check: ___________ √ � 3x + 5 = 3

√ ��

3 ( 4 __ 3 ) + 5 3

√ � 4 + 5 3 2 + 5 3 7 3 ; ✗ no solution

34. √ �� 2 - 7x = 2x

( √ �� 2 - 7x ) 2 = (2x) 2

2 - 7x = 4 x 2 0 = 4 x 2 + 7x - 2 0 = (4x - 1) (x + 2) 4x - 1 = 0 or x + 2 = 0

x = 1 __ 4 or x = -2

Check: ____________ √ �� 2 - 7x = 2x

√ ��

2 - 7 ( 1 __ 4 ) 2 ( 1 __

4 )

√ �� 21 ___ 4 1 __

2

1 __ 2 1 __

2 ✓

____________ √ �� 2 - 7x = 2x √ �� 2 - 7(2) 2(2)

√ �� 2 - 14 4 √ �� -12 4 ✗;

1 __ 4 is the only solution.

35. x = √ �� 12 + x

(x) 2 = ( √ �� 12 + x ) 2

x 2 = 12 + x x 2 - x - 12 = 0 (x - 4) (x + 3) = 0x - 4 = 0 or x + 3 = 0 x = 4 or x = -3Check: ___________ x = √ �� 12 + x ___________ x = √ �� 12 + x 4 √ �� 12 + 4 -3 √ �� 12 - 3 4 √ �� 16 -3

√ � 9

4 4 ✓ -3 34 is the only solution.

36. 6 + √ �� x - 1 = 4 Check: _____________ 6 + √ �� x - 1 = 4 √ �� x - 1 = -2 6 + √ �� 5 - 1 4

( √ �� x - 1 ) 2 = (-2) 2 6 + √ � 4 4

x - 1 = 4 6 + 2 4 x = 5 8 4 ✗no solution

37. √ �� 6 - 3x + 2 = x √ �� 6 - 3x = x - 2

( √ �� 6 - 3x ) 2 = (x - 2) 2

6 - 3x = x 2 - 4x + 4 0 = x 2 - x - 2 0 = (x - 2) (x + 1) x - 2 = 0 or x + 1 = 0 x = 2 or x = -1Check: _______________ √ �� 6 - 3x + 2 = x √ �� 6 - 3(2) + 2 2 √ � 0 + 2 2 2 2 ✓

_______________ √ �� 6 - 3x + 2 = x √ �� 6 - 3(-1) + 2 -1

√ � 9 + 2 -1 3 + 2 -1 5 -1 ✗2 is the only solution.

38. √ �� x - 2 = 2 - x

( √ �� x - 2 ) 2 = (2 - x) 2

x - 2 = 4 - 4x + x 2 0 = x 2 - 5x + 6 0 = (x - 2) (x - 3) x - 2 = 0 or x - 3 = 0 x = 2 or x = 3Check: _____________ √ �� x - 2 = 2 - x √ �� 2 - 2 2 - 2 √ � 0 0 0 0 ✓ _____________ √ �� x - 2 = 2 - x √ �� 3 - 2 2 - 3 √ � 1 -1 1 -1 ✗2 is the only solution.


39. 10 + √ � x = 5 √ � x = -5 ( √ � x ) 2 = (-5) 2 x = 25Check: ___________ 10 + √ � x = 5 10 + √ �� 25 5 10 + 5 5 15 5 ✗no solution

40. A = 1 __ 2 ( b 1 + b 2 ) h

14 = 1 __ 2 (4 + 10) √ �� 2x + 3

2 = √ �� 2x + 3

(2) 2 = ( √ �� 2x + 3 ) 2

4 = 2x + 3 1 = 2x

1 __ 2 = x

Check: ______________

A = 1 __ 2 ( b 1 + b 2 ) h

14 1 __ 2 (4 + 10) √

�� 2 ( 1 __

2 ) + 3

14 7 √ � 4 14 7(2) 14 14 ✓

x = 1 __ 2 ; h = √

�� 2 ( 1 __

2 ) + 3 = 2 cm


41. √ � 3x = 12

( √ � 3x ) 2 = (12) 2

3x = 144 x = 48

42. 2 = √ �� -2x

(2) 2 = ( √ �� -2x ) 2

4 = -2x -2 = x

43. √ �� -a = 5 ( √ �� -a ) 2 = (5) 2 -a = 25 a = -25

44. 11 = √ � c (11) 2 = ( √ � c ) 2 121 = c

45. √ �� x - 7 = 8

( √ �� x - 7 ) 2 = (8) 2

x - 7 = 64 x = 71

46. √ � x - 4 = 0 √ � x = 4 ( √ � x ) 2 = (4) 2 x = 16

47. √ �� 1 - 3x = 5

( √ �� 1 - 3x ) 2 = (5) 2

1 - 3x = 25 -3x = 24 x = -8

48. √ �� 5x + 1 + 2 = 6 √ �� 5x + 1 = 4

( √ �� 5x + 1 ) 2 = (4) 2

5x + 1 = 16 x = 3

49. 5 √ � x = 30 √ � x = 6 ( √ � x ) 2 = (6) 2 x = 36

50. √ � 2x ____

2 = 4

√ � 2x = 8

( √ � 2x ) 2 = (8) 2

2x = 64 x = 32

51. 5 √ �� -x = 20 √ �� -x = 4 ( √ �� -x ) 2 = (4) 2 -x = 16 x = -16

52. 3 √ �� 3p = 9 √ �� 3p = 3

( √ �� 3p ) 2 = (3) 2

3p = 9 p = 3

53. √ �� 3x - 13 = √ �� x + 3

( √ �� 3x - 13 ) 2 = ( √ �� x + 3 )

2

3x - 13 = x + 3 2x = 16 x = 8

54. √ � x - √ �� 6 - x = 0 √ � x = √ �� 6 - x

( √ � x ) 2 = ( √ �� 6 - x ) 2

x = 6 -x 2x = 6 x = 3

55. √ �� x + 5 = √ �� 2x - 4

( √ �� x + 5 ) 2 = ( √ �� 2x - 4 )

2

x + 5 = 2x - 4 9 = x

56. √ �� 4x - 2 = √ �� 3x + 4

( √ �� 4x - 2 ) 2 = ( √ �� 3x + 4 )

2

4x - 2 = 3x + 4 x = 6

57. √ �� 5x - 6 = √ �� 16 - 6x

( √ �� 5x - 6 ) 2 = ( √ �� 16 - 6x )

2

5x - 6 = 16 - 6x 11x = 22 x = 2

58. √ �� 12x - 3 = √ �� 4x + 93

( √ �� 12x - 3 ) 2 = ( √ �� 4x + 93 )

2

12x - 3 = 4x + 93 8x = 96 x = 12

59. √ �� x + 6 = 1

( √ �� x + 6 ) 2 = (1) 2

x + 6 = 1 x = -5

60. -2 √ � x = 6 √ � x =-3 ( √ � x ) 2 = (-3) 2 x = 9Check: _________ -2 √ � x = 6 -2 √ � 9 6 -2(3) 6 -6 6 ✗no solution


61. x = √ �� 2x + 15

(x) 2 = ( √ �� 2x + 15 ) 2

x 2 = 2x + 15 x 2 - 2x - 15 = 0 (x - 5) (x + 3) = 0x - 5 = 0 or x + 3 = 0 x = 5 or x = -3Check: ____________ x = √ �� 2x + 15

5 √ �� 2(5) + 15

5 √ �� 25 5 5 ✓ ____________ x = √ �� 2x + 15

-3 √ �� 2(-3) + 15

-3 √ � 9 -3 3 ✗5 is the only solution.

62. √ � 6x + 9 = 2 Check: ___________ √ � 6x + 9 = 2

√ � 6x = -7 √ ��

6 ( 49 ___ 6 ) + 9 2

( √ � 6x ) 2 = (-7) 2 √ �� 49 + 9 2

6x = 49 7 + 9 2

x = 49 ___ 6 16 2 ✗

no solution

63. √ �� 4 - 3x = x

( √ �� 4 - 3x ) 2 = (x) 2

4 - 3x = x 2 0 = x 2 + 3x - 4 0 = (x + 4) (x - 1) x + 4 = 0 or x - 1 = 0x = -4 or x = 1Check: ___________ √ �� 4 - 3x = x

√ �� 4 - 3(-4) -4

√ �� 16 -4 4 -4 ✗ ___________ √ �� 4 - 3x = x

√ �� 4 - 3(1) 1

√ � 1 1 1 1 ✓1 is the only solution.

64. √ �� 5x + 4 = x - 4

( √ �� 5x + 4 ) 2 = (x - 4) 2

5x + 4 = x 2 - 8x + 16 0 = x 2 -13x + 12 0 = (x - 12) (x - 1)x - 12 = 0 or x - 1 = 0 x = 12 or x = 1Check: ______________ √ �� 5x + 4 = x - 4

√ �� 5(12) + 4 12 - 4

√ �� 64 8 8 8 ✓ ______________ √ �� 5x + 4 = x - 4

√ �� 5(1) + 4 1 - 4 √ � 9 -3 3 -3 ✗12 is the only solution.

65. √ �� 2x + 2 = 2x

( √ �� 2x + 2 ) 2 = (2x) 2

2x + 2 = 4 x 2 0 = 2 x 2 - x - 1 0 = (2x + 1) (x - 1) 2x + 1 = 0 or x - 1 =0

x = - 1 __ 2 or x = 1

Check: ____________ √ �� 2x + 2 = 2x ____________ √ �� 2x + 2 = 2x

√ ��

2 (- 1 __ 2 ) + 2 2 (- 1 __

2 ) √ �� 2(1) + 2 2(1)

√ �� -1 + 2 -1 √ � 4 2 √ � 1 -1 2 2✓ 1 -1✗ 1 is the only solution.

66. √ �� x + 3 + 10 = 7 √ �� x + 3 = -3

( √ �� x + 3 ) 2 = (-3) 2

x + 3 = 9 x = 6Check: ______________ √ �� x + 3 + 10 = 7 √ �� 6 + 3 + 10 7 √ � 9 + 10 7 3 + 10 7 13 7✗no solution

67. A = 1 __ 2 bh

60 = 1 __ 2 (10)( √ � x )

12 = √ � x (12) 2 = ( √ � x ) 2 144 = x; 12 in.

68. √ � 3x = 9; √ � 3x = 9

( √ � 3x ) 2 = (9) 2

3x = 81 x = 27

69. √ � x - 3 = 4 √ � x - 3 = 4 √ � x = 7 ( √ � x ) 2 = (7) 2 x = 49

70. √ �� x - 3 = 4 √ �� x - 3 = 4

( √ �� x - 3 ) 2 = (4) 2

x - 3 = 16 x = 19


71. x = √ �� x + 6 ; x = √ �� x + 6 (x) 2 = ( √ �� x + 6 )

2

x 2 = x + 6 x 2 - x - 6 = 0 (x - 3) (x + 2) = 0x - 3 = 0 or x + 2 = 0 x = 3 or x = -2Check: __________ x = √ �� x + 6 __________ x = √ �� x + 6 3 √ �� 3 + 6 -2 √ �� -2 + 6 3 √ � 9 -2 √ � 4 3 3✓ -2 2 ✗3 is the only solution.

72. P = 2(� + w) � = √ �� 9 + 7

18 = 2 (5 + √ �� x + 7 ) = √ �� 16

9 = 5 + √ �� x + 7 = 4 m 4 = √ �� x + 7

(4) 2 = ( √ �� x + 7 ) 2

16 = x + 7 9 = x Dimensions: 5 m by 4 m

73. P = 2(b + h)

8 = 2 ( √ �� x + 3 + 1)

4 = √ �� x + 3 + 1 3 = √ �� x + 3

(3) 2 = ( √ �� x + 3 ) 2

9 = x + 3 6 = xb = √ �� x + 3 = √ �� 6 + 3 = √ � 9 = 3 in.Dimensions: 3 in. by 1 in.

74. P = 2(b + h) 30 = 2 (3 √ � x + 2 √ � x ) 15 = 5 √ � x 3 = √ � x (3) 2 = ( √ � x ) 2 9 = xb = 3 √ � x = 3 √ � 9 = 9 cmh = 2 √ � x = 2 √ � 9 = 6 cmDimensions:9 cm by 6 cm

75a. v = √ �� 2Em ______ m

28 = √ �� 2E(0.14)

_________ 0.14

3.92 = √ �� 0.28E 3.92 = √ �� 0.28 √ � E

3.92 ______ √ �� 0.28

= √ � E

( 3.92 ______ √ �� 0.28

) 2 = ( √ � E )

2

15.3664 _______ 0.28

= E

E = 54.88 joules

b. v = √ �� 2Em ______ m

0 = √ �� 2Em ______ m

0 = √ �� 2Em

0 = √ �� 2m √ � E

√ �� 2m = 0 or √ � E = 0 √ �� 2m ≠ 0 since m ≠ 0; m is in the denominator.Then, E = 0 joules.

76. t = √ ��

d 2 ____ 216

1= √ ��

d 2 ____ 216

1= √ � d 2

_____ √ �� 216

√ �� 216 = d d = 14.70 mi

77. v = √ �� 2.5r 65 = √ �� 2.5r 65 = √ �� 2.5 √ � r 65 _____

√ �� 2.5 = √ � r

( 65 _____ √ �� 2.5

) 2 = ( √ � r ) 2

4225 _____ 2.5

= r

r = 1690 ft

78. Radical equations may have extraneous solutions.

79.

√ � x + √ � y = √ �� 81

6 √ � y = 24

�

�

In (2), 6 √ � y = 24

√ � y = 4

( √ � y ) 2 = (4) 2 y = 16 Subst. y = 16 into (1)

√ � x + √ �� 16 = √ �� 81 √ � x + 4 = 9 √ � x = 5 ( √ � x ) 2 = (5) 2 x = 25Therefore, x = 25 and y = 16.

80. always

81. Sometimes; for a = b = 2, the statement is true. For a = 2 and b = -2, the statement is false.

82. Sometimes; for the equation √ � 2x = √ �� x 2 - 3 , the value of x must be nonnegative in order for the left side to be defined, so the statement is true. For the equation √ �� 7 - x = 3, the solution is -2 and the statement is false.


83. Student B made an error going from 5 - x = x + 9 to 4 = 2x. The student should have added x to both sides and subtracted 9 from both sides to get -4 = 2x.

84. 3 m

85. x ≤ 0 since the square root is only defined for nonnegative values. k ≥ 0 since the value of the square root must be nonnegative.

86a. 42 mi _____ 1 hr

= 42(5280) ft

_________ 3600 s

= 61.6 ft/s

b. v = 8 √ � d 61.6 = 8 √ � d

61.6 2 = (8 √ � d ) 2

3794.56 = 64d

d ≈ 59.29 ft

87. A; check: ______________ √ �� 8 - 2x - 2 = 2

√ �� 8 - 2(-4) - 2 2 √ �� 16 - 2 2 4 - 2 2 2 2 ✓

88. J; √ �� x + 1 + 1 = 0 √ �� x + 1 = -1But the square root of any real, positive number is always positive.

89. C; check: ___________ x = √ �� 12 - x 3 √ �� 12 - 3 3 √ � 9 3 3 ✓

90. G; check: __________________ √ �� x + 13 = 5 √ �� x - 11

√ �� 12 + 13 5 √ �� 12 - 11 √ �� 25 5 √ � 1 5 5 ✓

91. A; check: ______________ √ �� 3x - 2 = x - 2

√ �� 3(1) - 2 1 - 2

√ � 1 -1 1 -1 ✗but 1 = -1


92. √ �� x + 3 = x + 1

( √ �� x + 3 ) 2 = (x + 1) 2

x + 3 = x 2 + 2x + 1 0 = x 2 + x - 2 0 = (x + 2) (x - 1) x + 2 = 0 or x - 1 = 0 x = -2 or x = 1Check: _____________ √ �� x + 3 = x + 1 _____________ √ �� x + 3 = x + 1

√ �� -2 + 3 -2 + 1 √ �� 1 + 3 1 + 1 √ � 1 -1 √ � 4 2 1 -1 ✗ 2 2 ✓1 is the only solution.

93. √ �� x - 1 = x - 1

( √ �� x - 1 ) 2 = (x - 1) 2

x - 1 = x 2 - 2x + 1 0 = x 2 - 3x + 2 0 = (x - 2) (x - 1) x - 2 = 0 or x - 1 = 0 x = 2 or x = 1Check: _____________ √ �� x - 1 = x - 1 _____________ √ �� x - 1 = x - 1 √ �� 2 - 1 2 - 1 √ �� 1 - 1 1 - 1 √ � 1 1 √ � 0 0 1 1 ✓ 0 0 ✓1, 2 are both possible solutions.

94. x - 1 = √ �� 2x + 6

(x - 1) 2 = ( √ �� 2x + 6 ) 2

x 2 - 2x + 1 = 2x + 6 x 2 - 4x - 5 = 0 (x - 5) (x + 1) = 0x - 5 = 0 or x + 1 = 0 x = 5 or x = -1Check: ______________ x - 1 = √ �� 2x + 6 ______________ x - 1 = √ �� 2x + 6

5 - 1 √ �� 2(5) + 6 -1 - 1 √ �� 2(-1) + 6

4 √ �� 16 -2 √ � 4 4 4 ✓ -2 2 ✗5 is the only possible solution.

95. √ �� x 2 + 5x + 11 = x + 3

( √ �� x 2 + 5x + 11 ) 2 = (x + 3) 2

x 2 + 5x + 11 = x 2 + 6x + 9 2 = x

96. √ �� x 2 + 9x + 14 = x + 4

( √ �� x 2 + 9x + 14 ) 2 = (x + 4) 2

x 2 + 9x + 14 = x 2 + 8x + 16 x = 2

97. x + 2 = √ �� x 2 + 5x + 4

(x + 2) 2 = ( √ �� x 2 + 5x + 4 ) 2

x 2 + 4x + 4 = x 2 + 5x + 4 0 = x

98a.

b. The equation has no solution. This is clear from the graphs since they do not intersect.


99a.

b. The solution is x = 3, which is where the graphs intersect.

100. y = 4 _______ √ �� x - 2

√ �� x - 2 > 0

( √ �� x - 2 ) 2 > (0) 2

x - 2 > 0 x > 2x > 2; x cannot equal 2 because the denom. cannot equal 0.


101. 3.2 ___ 2.5

= x ___ 40

128 = 2.5x51.2 mi = x

102. 12.5 ____ x = 1 ___ 48

x = 12.5(48) x = 600 in = 50 ft

103. Number of PINs = 10 × 10 × 10 × 10 = 10,000

104. Number of samplers = 6 C 4 = 15

105. 106.

107.

MULTI-STEP TEST PREP, PAGE 830

1. C = πd = 3.14(135) = 423.9 m

2. r = d __ 2

= 135 ____ 2

= 67.5 mv = √ �� 0.001r = √ �� 0.001(67.5)

= √ �� 0.0675 ≈ 0.26 m/s

3. t = 30 min = 1800 s

v = d __ t

= 423.9 _____ 1800

= 0.24 m/s

4. Differences are due to rounding.

5. 135 m; the required distance is the diameter of the wheel.

6. When the passenger is at point P, the distance d is the hypotenuse of the right triangle shown, so d 2 = 2 r 2 by the Pythagorean Theorem;

d = √ �� 2 r 2

= √ �� 2 ( 67.5 2 ) ≈ 95.46 m

READY TO GO ON? PAGE 831

1. D = 113 √ � h = 113 √ �� 0.3 = 61.9 km

2. y = √ �� 3x - 7 3x ≥ 0 x ≥ 0Domain: { x | x ≥ 0 }

3. y = √ �� x - 5 x - 5 ≥ 0 x ≥ 5Domain: { x | x ≥ 5 }

4. y = √ �� 2x - 6 2x - 6 ≥ 0 2x ≥ 6 x ≥ 3Domain: { x | x ≥ 3 }

5. 6.

7. 8. √ �� 75 = √ �� 25(3) = √ �� 25 √ � 3 = 5 √ � 3

9. √ �� 300 ____ 3 = √ �� 100

= 10

10. √ �� a 2 b 3 = √ �� a 2 b 2 b

= √ �� a 2 b 2 √ � b = ab √ � b

11. √ �� 98x y 2 = √ �� 49 y 2 (2x)

= √ �� 49 √ � y 2 √ � 2x

= 7y √ � 2x

12. √ �� 32 ___ 25

= √ ��

16(2)

_____ 25

= √ �� 16 √ � 2

_______ √ �� 25

= 4 √ � 2 ____ 5

13. √ �� 128 ____ 121

= √ ��

64(2)

_____ 121

= √ �� 64 √ � 2

_______ √ �� 121

= 8 √ � 2 ____ 11

14. √ ��

4 b 2 ___ 81

= √ �� 4 b 2

_____ √ �� 81

= 2b ___ 9

15. √ ��

75 a 9 ____ 49 a 3

= √

��

25(3) a 6 _______

49

= √ �� 25 a 6 (3)

________ √ �� 49

= √ �� 25 a 6 √ � 3

_________ √ �� 49

= 5 a 3 √ � 3

______ 7


16. diagonal = √ �� 19.2 2 + 14.4 2 = √ �� 368.64 + 207.36 = √ �� 576 = 24 in.

17. 12 √ � 7 - 5 √ � 7 = 7 √ � 7 18. 3 √ � x + 3 √ � x = 6 √ � x

19. √ �� 12 + √ �� 75 = √ �� 4(3) + √ �� 25(3) = √ � 4 √ � 3 + √ �� 25 √ � 3 = 2 √ � 3 + 5 √ � 3 = 7 √ � 3

20. 5 √ �� 50 + √ �� 98 = 5 √ �� 25(2) + √ �� 49(2) = 5 √ �� 25 √ � 2 + √ �� 49 √ � 2 = 5(5) √ � 2 + 7 √ � 2 = 25 √ � 2 + 7 √ � 2 = 32 √ � 2

21. 4 √ � 3 - 3 √ � 4 = 4 √ � 3 - 3(2) = 4 √ � 3 - 6

22. √ �� 98x + √ �� 18x - √ �� 200x = √ �� 49(2x) + √ �� 9(2x) - √ �� 100(2x)

= √ �� 49 √ � 2x + √ � 9 √ � 2x -10 √ � 2x

= 7 √ � 2x + 3 √ � 2x - 10 √ � 2x = 0

23. √ � 6 √ � 11 = √ �� 6(11) = √ �� 66

24. √ � 3 √ � 8 = √ �� 3(8) = √ �� 24 = √ �� 4(6)

= √ � 4 √ � 6 = 2 √ � 6

25. 4 √ �� 12x √ � 3x = 4 √ �� 12x(3x)

= 4 √ �� 36 x 2 = 4(6x) = 24x

26. (3 - √ � 3 ) (5 + √ � 3 ) = 15 + 3 √ � 3 - 5 √ � 3 - √ � 3 √ � 3 = 15 - 2 √ � 3 - 3 = 12 - 2 √ � 3

27. √ �� 19

____ √ � 3

= √ �� 19

____ √ � 3

( √ � 3

___ √ � 3

)

= √ �� 19(3)

______ 3

= √ �� 57

____ 3

28. √ �� 14 ____ √ � 8

= √ �� 14 ___ 8

= √ � 7 __ 4

= √ � 7 ___ √ � 4

= √ � 7 ___ 2

29. √ �� 6b

____ √ � 8

= √ �� 6b ___ 8

= √ �� 3b ___ 4

= √ �� 3b

____ √ � 4

= √ �� 3b

_ 2

30. √ �� 27 ____ √ � 3t

= √ �� 27 ____ √ � 3t

( √ � 3t

____ √ � 3t

)

= √ �� 27(3t)

_______ 3t

= √ �� 81t

_____ 3t

= 9 √ � t

____ 3t

= 3 √ � t

____ t

31. √ � x - 4 = 21 √ � x = 25 ( √ � x ) 2 = (25) 2 x = 625

32. -3 √ � x = -12 √ � x = 4 ( √ � x ) 2 = (4) 2 x = 16

33. 5 √ � x ____ 2 = 40

√ � x = 16 ( √ � x ) 2 = (16) 2 x = 256

34. √ �� 4x - 2 - √ �� 43 - x = 0 √ �� 4x - 2 = √ �� 43 - x

( √ �� 4x - 2 ) 2 = ( √ �� 43 - x )

2

4x - 2 = 43 - x 5x = 45 x = 9

35. √ �� 20 + x = x

( √ �� 20 + x ) 2 = (x) 2

20 + x = x 2 0 = x 2 - x - 20 0 = (x - 5) (x + 4) x - 5 = 0 or x + 4 = 0 x = 5 or x = -4Check: ___________ √ �� 20 + x = x ___________ √ �� 20 + x = x √ �� 20 + 5 5 √ �� 20 - 4 -4 √ �� 25 5 √ �� 16 -4 5 5 ✓ 4 -4 ✗5 is the only solution.

36. √ � 4x + 12 = 10 √ � 4x = -2

( √ � 4x ) 2 = (-2) 2

4x = 4 x = 1

Check: _____________ √ � 4x + 12 = 10 √ �� 4(1) + 12 10 2 + 12 10 14 10 ✗no solution


STUDY GUIDE: REVIEW, PAGES 836−839

EXERCISES, PAGE 836

1. square-root function 2. exponential decay

3. common ratio 4. exponential function

LESSON 11-1, PAGE 836

5. 3 _ 1 = 3; 9 _

3 = 3; 27 _

9 = 3

The next three terms are: 27 · 3 = 81 81 · 3 = 243 243 · 3 = 729

6. -6 ___ 3 = -2; 12 ___ -6

= -2; -24 ____ 12

= -2

The next three terms are: (-24) (-2) = 48 48 (-2) = -96 (-96) (-2) = 192

7. 40 ___ 80

= 1 __ 2 ; 20 ___

40 = 1 __

2 ; 10 ___

20 = 1 __

2

The next three terms are: 10 ( 1 __ 2 ) = 5

5 ( 1 __ 2 ) = 2.5

2.5 ( 1 __ 2 ) = 1.25

8. -4 ___ -1 = 4; -16 ____ -4

= 4; -64 ____ -16 = 4

The next three terms are: (-64) (4) = -256 (-256) (4) = -1024 (-1024) (4) = -4096

9. a n = a 1 r n - 1

a 10 = (4) ( 5 10 - 1 ) = (4) (1,953,125) = 7,812,500

10. 12 ___ 4 = 3; 36 ___

12 = 3; 108 ____

36 = 3

Hence, r = 3 and a 1 = 4 a n = a 1 r n - 1

a 15 = (4) ( 3 15 - 1 ) = (4) (4,782,969) = 19,131,876




13. 14.


15. y = a (1 + r) t y = 9 (1 + 0.15) t y = 9 (1.15) t ;y = 9 (1.15) 7 y = 24

16. y = a (1 - r) t y = 24,500 (1 - 0.04) t y = 24,500 (0.96) t ;y = 24,500 (0.96) 50 y = 3182


17.

quadratic

18.

linear

19.

exponential

20. exponential

21. quadratic

22. linear

23. y = 1.5x; 15 h


24. � = √ � S __ 6

= √ �� 135 ____ 6

= √ �� 22.5 = 4.74 cm

25. y = √ � x + 5x ≥ 0Domain: { x | x ≥ 0 }

26. y = √ �� x + 4 x + 4 ≥ 0 x ≥ -4Domain: { x | x ≥ -4 }

27. y = 8 - √ � 3x 3x ≥ 0 x ≥ 0Domain: { x | x ≥ 0 }

28. y = 2 √ �� x + 2 x + 2 ≥ 0 x ≥ -2Domain: { x | x ≥ -2 }

29. y = 1 + √ �� 3x - 4 3x - 4 ≥ 0 3x ≥ 4

x ≥ 4 __ 3

Domain: { x | x ≥ 4 __ 3

}

30. y = √ �� 2x + 6 2x + 6 ≥ 0 2x ≥ -6 x ≥ -3Domain: { x | x ≥ -3 }

31. y = √ �� 2x - 7 2x - 7 ≥ 0 2x ≥ 7

x ≥ 7 __ 2

Domain: { x | x ≥ 7 __ 2

}


32. y = √ �� 5x + 18 5x + 18 ≥ 0 5x ≥ -18

x ≥ - 18 ___ 5

Domain: { x | x ≥ - 18 ___ 5 }

33. y = √ �� 4x - 3 4x - 3 ≥ 0 4x ≥ 3

x ≥ 3 __ 4

Domain: { x | x ≥ 3 __ 4 }

34. y = 3 √ �� x - 1 x - 1 ≥ 0 x ≥ 1Domain: { x | x ≥ 1 }

35.

36. 37.

38. 39.

40. 41.

42. 43.

44.


45. √ �� 121 = 11 46. √ � n 4 = n 2

47. √ �� (x + 3) 2 = x + 3 48. √ �� 75 ___ 3 = √ �� 25

= 5

49. √ �� 36 d 2 = √ �� 36 √ � d 2 = 6d

50. √ �� y 6 x = √ � y 6 √ � x = y 3 √ � x

51. √ �� 12 = √ �� 3(4) = √ � 3 √ � 4 = 2 √ � 3

52. √ �� 32a b 5 = √ �� (16)(2)a b 4 (b)

= √ �� 16 √ �� 2ab √ � b 4 = 4 b 2 √ �� 2ab

53. √ � 5 __ 4 =

√ � 5 ___

√ � 4

= √ � 5

___ 2

54. √ ��

t 3 ____ 100t

= √ ��

t 2 ____ 100

= √ � t 2 _____

√ �� 100

= t ___ 10

55. √ �� 8 ___ 18

= √ � 4 __ 9

= √ � 4 ___ √ � 9

= 2 __ 3

56. √

��

32 p 4 ____

49 =

√ ��

(16)(2) p 4

________ 49

= √ �� 16 √ � 2 √ � p 4

__________ √ �� 49

= 4 p 2 √ � 2

______ 7

57. √ ��

s 2 t 9 ____ s 4

= √ ��

t 8 t __

s 2

= √ � t 8 √ � t

______ √ � s 2

= t 4 √ � t

____ s

58. √ ��

72 b 8 ____ 225

= √

��

4(2) b 8 ______

25

= √ � 4 √ � 2 √ � b 8

_________ √ �� 25

= 2 b 4 √ � 2 ______ 5


59. 6 √ � 7 + 3 √ � 7 = 9 √ � 7 60. 4 √ � 3 - √ � 3 = 3 √ � 3

61. 3 √ � 2 + 2 √ � 3 = 3 √ � 2 + 2 √ � 3

62. 9 √ � 5t - 8 √ � 5t = √ � 5t

63. √ �� 50 - √ �� 18 = √ �� (25)(2) - √ �� (9)(2) = √ �� 25 √ � 2 - √ � 9 √ � 2 = 5 √ � 2 - 3 √ � 2 = 2 √ � 2

64. √ �� 12 + √ �� 20 = √ �� (4)(3) + √ �� (4)(5) = √ � 4 √ � 3 + √ � 4 √ � 5 = 2 √ � 3 + 2 √ � 5

65. √ �� 20x - √ �� 80x = √ �� (4)(5x) - √ �� (16)(5x) = √ � 4 √ � 5x - √ �� 16 √ � 5x = 2 √ � 5x - 4 √ � 5x = -2 √ � 5x


66. 4 √ �� 54 - √ �� 24 = 4 √ �� (9)(6) - √ �� (4)(6) = 4 √ � 9 √ � 6 - √ � 4 √ � 6 = 4(3) √ � 6 - 2 √ � 6 = 10 √ � 6


67. √ � 2 √ � 7 = √ �� (2)(7) = √ �� 14

68. √ � 3 √ � 6 = √ �� (3)(6) = √ �� 18 = √ �� 9(2)

= 3 √ � 2

69. 3 √ � 2x √ �� 14 = 3 √ �� (2x)(14) = 3 √ �� 28x = 3 √ �� (4)(7x)

= 6 √ � 7x

70. (5 √ � 6 ) 2 = (5 √ � 6 ) (5 √ � 6 )

= 25(6) = 150

71. √ � 2 (4 - √ � 8 ) = 4 √ � 2 - √ � 2 √ � 8 = 4 √ � 2 - √ �� (2)(8)

= 4 √ � 2 - √ �� 16 = 4 √ � 2 - 4

72. (8 + √ � 7 ) 2 = (8 + √ � 7 ) (8 + √ � 7 )

= 64 + 8 √ � 7 + 8 √ � 7 + 7 = 71 + 16 √ � 7

73. 4 ___ √ � 5

= ( 4 ___ √ � 5

) ( √ � 5

___ √ � 5

)

= 4 √ � 5

____ 5

74. a √ � 9 ____

√ � 2 = 3a ___

√ � 2

= ( 3a ___ √ � 2

) ( √ � 2 ___ √ � 2

)

= 3a √ � 2 _____ 2

75. √ � 8

____ 2 √ � 6

= ( √ � 8

____ 2 √ � 6

) ( √ � 6

___ √ � 6

)

= √ �� 8(6)

_____ 2(6)

= √ �� 48

____ 12

= √ �� 16(3)

______ 12

= 4 √ � 3

____ 12

= √ � 3

___ 3

76. √ � 5

____ √ �� 2n

= ( √ � 5

____ √ �� 2n

) ( √ �� 2n ____ √ �� 2n

)

= √ �� 10n

_____ 2n

77. √ �� 18

____ √ �� 12

= √ �� (2)(9)

_______ √ �� (4)(3)

= 3 √ � 2 ____ 2 √ � 3

= ( 3 √ � 2 ____ 2 √ � 3

) ( √ � 3

___ √ � 3

)

= 3 √ �� 2(3)

______ 2(3)

= 3 √ � 6

_ 6

= √ � 6

___ 2

78. -3 ___ √ � 3

= ( -3 ___ √ � 3

) ( √ � 3

___ √ � 3

)

= -3 √ � 3

______ 3

= - √ � 3


79. √ � x = 8 ( √ � x ) 2 = (8) 2 x = 64

80. √ � 2x = 4

( √ � 2x ) 2 = (4) 2

2x = 16 x = 8

81. √ �� x + 6 = 3

( √ �� x + 6 ) 2 = (3) 2

x + 6 = 9 x = 3

82. -3 √ � x = -15 √ � x = 5 ( √ � x ) 2 = (5) 2 x = 25

83. 3 √ �� -x = 27 √ �� -x = 9 ( √ �� -x ) 2 = (9) 2 -x = 81 x = -81

84. 4 √ � x ____ 5 = 8

√ � x = 10 ( √ � x ) 2 = (10) 2 x = 100

85. √ �� x + 1 = √ �� 3x - 5

( √ �� x + 1 ) 2 = ( √ �� 3x - 5 )

2

x + 1 = 3x - 5 6 = 2x 3 = x

86. √ �� x - 2 + 4 = 3 √ �� x - 2 = -1

( √ �� x - 2 ) 2 = (-1) 2

x - 2 = 1 x = 3Check: _____________ √ �� x - 2 + 4 = 3 √ �� 3 - 2 + 4 3 √ � 1 + 4 3 5 3 ✗no solution

87. 12 = 4 √ �� 2x + 1 3 = √ �� 2x + 1

(3) 2 = ( √ �� 2x + 1 ) 2

9 = 2x + 1 4 = x


88. √ �� x - 5 = √ �� 7 - x

( √ �� x - 5 ) 2 = ( √ �� 7 - x )

2

x - 5 = 7 - x 2x = 12 x = 6

89. √ �� x + 2 = 3

( √ �� x + 2 ) 2 = (3) 2

x + 2 = 9 x = 7

90. √ �� 2x - 3 = 4

( √ �� 2x - 3 ) 2 = (4) 2

2x - 3 = 16 2x = 19

x = 19 ___ 2

91. 4 √ �� x - 3 = 12

( √ �� x - 3 ) 2 = (3) 2

x - 3 = 9 x = 12

92. √ �� x + 6 = x

( √ �� x + 6 ) 2 = (x) 2

x + 6 = x 2 0 = x 2 - x - 6 0 = (x - 3) (x + 2) x - 3 = 0 or x + 2 = 0 x = 3 or x = -2

So, x = 3

Check: __________ √ �� x + 6 = x √ �� 3 + 6 3 √ � 9 3 3 3 ✓ __________ √ �� x + 6 = x √ �� -2 + 6 -2 √ � 4 -2 2 -2 ✗

93. √ �� 3x + 4 = x

( √ �� 3x + 4 ) 2 = (x) 2

3x + 4 = x 2 0 = x 2 - 3x - 4 0 = (x - 4) (x + 1) x - 4 = 0 or x + 1 = 0 x = 4 or x = -1Check: ___________ √ �� 3x + 4 = x

√ �� 3(4) + 4 4

√ �� 16 4 4 4 ✓ ___________ √ �� 3x + 4 = x

√ �� 3(-1) + 4 -1

√ � 1 -1 1 -1 ✗So, x = 4

94. √ �� 2x + 6 = x - 1

( √ �� 2x + 6 ) 2 = (x - 1) 2

2x + 6 = x 2 - 2x + 1 0 = x 2 - 4x - 5 0 = (x - 5) (x + 1) x - 5 = 0 or x + 1 = 0 x = 5 or x = -1Check: ______________ √ �� 2x + 6 = x - 1

√ �� 2(5) + 6 5 - 1

√ �� 16 4 4 4 ✓ ______________ √ �� 2x + 6 = x - 1

√ �� 2(-1) + 6 -1 - 1

√ � 4 -2 2 -2 ✗So, x = 5

CHAPTER TEST, PAGE 840

1. 6 __ 2 = 3; 18 ___

6 = 3; 54 ___

18 = 3

The next three terms are: 54(3) = 162 162(3) = 486 486(3) = 1458

2. 2400 _____ 4800

= 1 __ 2 ; 1200 _____

2400 = 1 __

2 ; 600 _____

1200 = 1 __

2

The next three terms are: 600 ( 1 __ 2 ) = 300

300 ( 1 __ 2 ) = 150

150 ( 1 __ 2 ) = 75

3. 20 ___ -4 = -5; -100 _____

20 = -5; 500 _____ -100

= -5

The next three terms are: 500 (-5) = -2500 (-2500) (-5) = 12,500 12,500 (-5) = -62500

4. a 1 = 2; r = 2 a n = a 1 r n - 1 a 7 = (2) 2 6 = 128

5.

6. 7.


8. 9. f(x) = 3 (1.25) x f(5) = 3 (1.25) 5 ≈ 9 cm

10. A = P (1 + r __ n ) nt

A = 5600 (1 + 0.036 _____ 4 )

4t

A = 5600 (1.009) 4t

;A = 5600 (1.009) 4(6) = $6943.46

11. y = a (1 - r) t y = 24,000 (1 - 0.05) t y = 24,000 (0.95) t ;y = 24,000 (0.95) 15 ≈ 11,119 trees

12. linear; y = 2x + 3 13. exponential; r = 3

14. The bacteria population is tripling every hour; y = 6 (3) x ;y = 6 (3) 10 = 354,294

15. y = 6 + √ � x x ≥ 0Domain: { x | x ≥ 0 }

16. y = -2 √ �� x + 9 x + 9 ≥ 0 x ≥ -9Domain: { x | x ≥ -9 }

17. y = x + √ �� 3x - 3 3x - 3 ≥ 0 3x ≥ 3 x ≥ 1Domain: { x | x ≥ 1 }

18.

19. 20.

21. √ �� 27 = √ �� (9) (3) = √ � 9 √ � 3 = 3 √ � 3

22. √ �� 75 m 4 = √ �� (25) (3) ( m 4 )

= √ �� 25 √ � 3 √ �� m 4 = 5 m 2 √ � 3

23. √ ��

x 6 __ y 2

= √ � x 6 ____ √ � y 2

= x 3 __ y

24. √ ��

p 9 _____

144p =

√ ��

p 8 ____

144

= √ � p 8

_____ √ �� 144

= p 4

___ 12

25. 4 √ �� 10 - 2 √ �� 10 = 2 √ �� 10

26. 5 √ � 3y + √ � 3y = 6 √ � 3y

27. √ � 8 - √ �� 50 = √ �� 4 (2) - √ �� (25) (2) = √ � 4 √ � 2 - √ �� 25 √ � 2 = 2 √ � 2 - 5 √ � 2 = -3 √ � 2

28. 2 √ �� 75 - √ �� 32 + √ �� 48

= 2 √ �� (25) (3) - √ �� (16) 2 + √ �� (16) (3) = 2 √ �� 25 √ � 3 - √ �� 16 √ � 2 + √ �� 16 √ � 3 = 10 √ � 3 - 4 √ � 2 + 4 √ � 3 = 14 √ � 3 - 4 √ � 2

29. √ � 2 √ �� 3m = √ �� 6m 30. √ �� 128d

______ √ � 5

= √ �� (64) (2d)

_________ √ � 5

= 8 √ �� 2d

_____ √ � 5

= ( 8 √ �� 2d

_____ √ � 5

) ( √ � 5

___ √ � 5

)

= 8 √ �� 10d

______ 5

31. √ � 3 ( √ �� 21 - 2) = √ �� 3 (21) - 2 √ � 3 = √ �� 63 - 2 √ � 3

= √ �� (9) (7) - 2 √ � 3 = 3 √ � 7 - 2 √ � 3

32. ( √ � 3 - 2) ( √ � 3 + 4) = 3 + 4 √ � 3 - 2 √ � 3 - 8 = 2 √ � 3 - 5

33. √ � 2x = 6

( √ � 2x ) 2 = (6) 2

2x = 36 x = 18

34. √ �� 3x + 4 - 2 = 5 √ �� 3x + 4 = 7

( √ �� 3x + 4 ) 2 = (7) 2

3x + 4 = 49 3x = 45 x = 15

35. 2 √ � x ____ 3 = 8

√ � x = 12

( √ � x ) 2 = (12) 2 x = 144

36. √ �� 5x + 1 = √ �� 2x - 2

( √ �� 5x + 1 ) 2 = ( √ �� 2x - 2 )

2

5x + 1 = 2x - 2 3x = -3 x = -1Check: _________________ √ �� 5x + 1 = √ �� 2x - 2

√ �� 5(-1) + 1 √ �� 2(-1) - 2

√ �� -4 √ �� -4 ✗

Since √ �� -4 is undefined, no solution.


COLLEGE ENTRANCE EXAM PRACTICE, PAGE 841

1. D; x - 4 ≥ 0x ≥ 4

2. E; √ � 8 √ � 3

______ √ � 5

= √ �� 24 ____ √ � 5

= ( √ �� 24 ____ √ � 5

) ( √ � 5

___ √ � 5

)

= √ �� 4(6)(5)

________ 5

= 2 √ �� 30

_____ 5

3. C; Check: ___________

√ �� 6 - 3x

________ 5 = 3

√ �� 6 - 3(-73)

___________ 5 3

√ �� 6 + 219

_________ 5 3

√ �� 225

_____ 5 3

15 ___ 5 3

3 3 ✓

4. D; 512 ____ 32

= r 2

r = 4 a 8 = a 5 r 3 = 512 (4) 3 = 32,768

5. B


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