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Chapter 9 Review Exercises

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Chapter 9 Review Exercises

Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

Exercise 1

64

Exercise 2

144

Answer

12

Exercise 3

25

Exercise 4

81

Answer

−9

Exercise 5

9

Exercise 6

36

Answer

not a real number

Exercise 7

64+225

Exercise 8

64+225

Answer

17

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

Exercise 9

28

Exercise 10

155

Answer

12<155<13

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

Exercise 11

15

Example 12

57

Answer

7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Exercise 13

q2

Exercise 14

64b2

Answer

8b

Exercis\e 15

121a2

Exercise 16

225m2n2

Answer

15mn

Exercise 17

100q2

Exercise 18

49y2

Answer

7y

Exercise 19

4a2b2

Exercise 20

121c2d2

Answer

11cd

Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Exercise 21

300

Exercise 22

98

Answer

72

Exercise 23

x13

Exercise 24

y19

Answer

y9y

Exercise 25

16m4

Exercise 26

36n13

Answer

6n6n

Exercise 27

288m21

Exercise 28

150n7

Answer

5n36n

Exercise 29

48r5s4

Exercise 30

108r5s3

Answer

6r2s3rs

Exercise 31

10505

Exercise 32

6+726

Answer

1+2

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Exercise 33

1625

Exercise 34

8136

Answer

32

Exercise 35

x8x4

Exercise 36

y6y2

Answer

y2

Exercise 37

98p62p2

Exercise 38

72q82q4

Answer

6q2

Exercise 39

65121

Exercise 40

26169

Answer

2613

Exercise 41

64x425x2

Exercise 42

36r1016r5

Answer

3r2r2

Exercise 43

48p3q527pq

Exercise 44

12r5s775r2s

Answer

2rs3r5

Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

Exercise 45

32+2

Exercise 46

55+75

Answer

125

Exercise 47

4y+4y

Exercise 48

6m2m

Answer

4m

Exercise 49

37+277

Exercise 50

813+23+313

Answer

1113+23

Exercise 51

35xy5xy+35xy

Exercise 52

23rs+3rs5rs

Answer

33rs5rs

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

Exercise 53

32+32

Exercise 54

8+32

Answer

52

Exercise 55

72+50

Exercise 56

48+75

Answer

93

Exercise 57

332+98

Exercise 58

132718192

Answer

0

Exercise 59

50y572y5

Exercise 60

Add exercises text here.

Answer

17n22

Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

Exercise 61

2·20

Exercise 62

22·614

Answer

247

Exercise 63

2m2·20m4

Exercise 64

(62y)(350y3)

Answer

180y2

Exercise 65

(63v4)(530v)

Exercise 66

(8)2

Answer

8

Exercise 67

(10)2

Exercise 68

(25)(55)

Answer

50

Exercise 69

(33)(518)

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

Exercise 70

10(27)

Answer

20107

Exercise 71

3(4+12)

Exercise 72

(5+2)(32)

Answer

1322

Exercise 73

(537)(127)

Exercise 74

(13x)(5+2x)

Answer

513x6x

Exercise 75

(3+4y)(10y)

Exercise 76

(1+6p)2

Answer

1+12p+36p

Exercise 77

(265)2

Exercise 78

(3+27)(327)

Answer

−19

Exercise 79

(611)(6+11)

Divide Square Roots

Divide Square Roots

In the following exercises, simplify.

Exercise 80

7510

Answer

32

Exercise 81

2126

Exercise 82

4827

Answer

43

Exercise 83

75x73x3

Exercise 84

20y52y

Answer

y210

Exercise 85

98p6q42p4q8

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Exercise 86

1015

Answer

2153

Exercise 87

66

Exercise 88

535

Answer

53

Exercise 89

1026

Exercise 90

328

Answer

2114

Exercise 91

975

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

Exercise 92

44+27

Answer

1612311

Exercise 93

5210

Exercise 94

425

Answer

845

Exercise 95

548

Exercise 96

2p+3

Answer

2p6p3

Exercise 97

x2x+2

Solve Equations with Square Roots

Solve Radical Equations

In the following exercises, solve the equation.

Exercise 98

7z+1=6

Answer

5

Exercise 99

4u24=0

Exercise 100

6m+45=0

Answer

72

Exercise 101

2u3+2=0

Exercise 102

u4+4=u

Answer

no solution

Exercise 103

v9+9=0​​​​​​​

Exercise 104

r4r=10

Answer

13

Exercise 105

s9s=9

Exercise 106

22x74=8

Answer

432

Exercise 107

2x=2x7

Exercise 108

a+3=a+9

Answer

0

Exercise 109

r+3=r+4​​​​​​​

Exercise 110

u+2=u+5

Answer

116

Exercise 111

n+111=n+4

Exercise 112

y+5+1=2y+3

Answer

11

​​​​​​​Use Square Roots in Applications

In the following exercises, solve. Round approximations to one decimal place.

Exercise 113

A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula s=A to find the length of each side of his lawn.​​​​​​​

Exercise 114

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula t=h4 to find how many seconds it took for the package to reach the hiker.

Answer

7.5 seconds

Exercise 115

Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula s=24d to find the speed of the car before the brakes were applied.

Higher Roots

Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise 116
  1. 664
  2. 364
Answer
  1. 2
  2. 4
Exercise 117
  1. 327
  2. 464
Exercise 118
  1. 9d9
  2. 8v8
Answer
  1. d
  2. |v|
Exercise 119
  1. 5a10
  2. 3b27
Exercise 120
  1. 416x8
  2. 664y12
Answer
  1. 2x2
  2. 2y2
Exercise 121
  1. 7128r14
  2. 481s24

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise 122
  1. 9d9

.

Answer
  1. d

.​​​​​​​​​​​​​​

Exercise 123
  1. 354
  2. 4128
Exercise 124
  1. 564c8
  2. 448d7
Answer
  1. 2c52c3
  2. 2d43d3
Exercise 125
  1. 3343q7
  2. 6192r9
Exercise 126
  1. 3500
  2. 416
Answer
  1. 534
  2. not a real number
​​​​​​​Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise 127

5r10r5

Exercise 128

3w12w2

Answer

w33w

Exercise 129

464y84y5

Exercise 130

354z92z3

Answer

3z2

Exercise 131

664a7b2

​​​​​​​Add and Subtract Higher Roots

In the following exercises, simplify.

Exercise 132

45202520

Answer

2520

Exercise 133

4318+3318

Exercise 134

412504162

Answer

242

Exercise 135

3640c5380c3

Exercise \PageIndex{136}

\sqrt[5]{96t^8}+\sqrt[5]{486t^4}

Answer

2t^\sqrt[5]{3t^3}+3\sqrt[5]{2t^4}

Rational Exponents

Simplify Expressions with a^{\frac{1}{n}}

In the following exercises, write as a radical expression.

Exercise \PageIndex{137}

r^{\frac{1}{8}}

Exercise \PageIndex{138}

s^{\frac{1}{10}}

Answer

.

​​​​​​​In the following exercises, write with a rational exponent.

Exercise \PageIndex{139}

\sqrt[5]{u}

Exercise \PageIndex{140}

\sqrt[6]{v}

Answer

v^{\frac{1}{6}}

Exercise \PageIndex{141}

\sqrt[3]{9m}

Exercise \PageIndex{142}

\sqrt[6]{10z}

Answer

(10z)^{\frac{1}{6}}

​​​​​​​In the following exercises, simplify.

Exercise \PageIndex{143}

16^{\frac{1}{4}}

Exercise \PageIndex{144}

32^{\frac{1}{5}}

Answer

2

Exercise \PageIndex{145}

(−125)^{\frac{1}{3}}

Exercise \PageIndex{146}

(125)^{−\frac{1}{3}}

Answer

\frac{1}{5}

Exercise \PageIndex{147}

(−9)^{\frac{1}{2}}

Exercise \PageIndex{148}

(36)^{−\frac{1}{2}}

Answer

\frac{1}{6}

​​​​​​​Simplify Expressions with a^{\frac{m}{n}}

In the following exercises, write with a rational exponent.

Exercise \PageIndex{149}

\sqrt[3]{q^5}

Exercise \PageIndex{150}

\sqrt[5]{n^8}

Answer

n^{\frac{8}{5}}

In the following exercises, simplify.

Exercise \PageIndex{151}

27^{−\frac{2}{3}}

Exercise \PageIndex{152}

64^{\frac{5}{2}}

Answer

32,768

Exercise \PageIndex{153}

36^{\frac{3}{2}}

Exercise \PageIndex{154}

81^{−\frac{5}{2}}

Answer

\frac{1}{59,049}

​​​​​​​Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Exercise \PageIndex{155}

3^{\frac{4}{5}}·3^{\frac{6}{5}}

Exercise \PageIndex{156}

(x^6)^{\frac{4}{3}}

Answer

x^8

Exercise \PageIndex{157}

\frac{z^{\frac{5}{2}}}{z^{\frac{7}{5}}}

Exercise \PageIndex{158}

(16s^{\frac{9}{4}})^{\frac{1}{4}}

Answer

2s^{\frac{9}{16}}

Exercise \PageIndex{159}

(m^{8}n^{12})^{\frac{1}{4}}

Exercise \PageIndex{160}

\frac{z^{\frac{2}{3}}·z^{−\frac{1}{3}}}{z^{−\frac{5}{3}}}

Answer

z^2​​​​​​​

Practice Test

In the following exercises, simplify.

Exercise \PageIndex{161}

\sqrt{81+144}

Exercise \PageIndex{162}

\sqrt{169m^{4}n^{2}}

Answer

13m^{2}|n|

Exercise \PageIndex{163}

\sqrt{36n^{13}}

Exercise \PageIndex{164}

3\sqrt{13}+5\sqrt{2}+\sqrt{13}

Answer

4\sqrt{13}+5\sqrt{2}

Exercise \PageIndex{165}

5\sqrt{20}+2\sqrt{125}

Exercise \PageIndex{166}

(3\sqrt{6y})(\sqrt{250y^3})

Answer

180y^2\sqrt{3}

Exercise \PageIndex{167}

(2−5\sqrt{x})(3+\sqrt{x})

Exercise \PageIndex{168}

(1−2\sqrt{q})^2

Answer

1−4\sqrt{q}+4q

Exercise \PageIndex{169}
  1. \sqrt{a^{12}}
  2. \sqrt[3]{b^{21}}
Exercise \PageIndex{170}
  1. \sqrt[4]{81x^{12}}
  2. \sqrt[6]{64y^{18}}
Answer
  1. 3x^3
  2. 2y^3
Exercise \PageIndex{171}

\sqrt[6]{\frac{64r^{12}}{25r^6}}​​​​​​​

Exercise \PageIndex{172}

\sqrt{\frac{14y^3}{7y}}

Answer

y\sqrt{2}

Exercise \PageIndex{173}

\frac{\sqrt{256x^7}}{\sqrt{54x^2}}

Exercise \PageIndex{174}

\sqrt[4]{512}−2\sqrt[4]{32}

Answer

0

Exercise \PageIndex{175}
  1. 256^{\frac{1}{4}}
  2. 243^{\frac{1}{5}}
Exercise \PageIndex{176}

49^{\frac{3}{2}}

Answer

343

Exercise \PageIndex{177}

25^{−\frac{5}{2}}

Exercise \PageIndex{178}

\frac{w^{\frac{3}{4}}}{w^{\frac{7}{4}}}

Answer

\frac{1}{w}

Exercise \PageIndex{179}

(27s^{\frac{3}{5}})^{\frac{1}{3}}

​​​​​​​In the following exercises, rationalize the denominator.

Exercise \PageIndex{180}

\frac{3}{2\sqrt{6}}

Answer

\frac{\sqrt{6}}{4}

Exercise \PageIndex{181}

\frac{\sqrt{3}}{\sqrt{x}+\sqrt{5}}

In the following exercises, solve.

Exercise \PageIndex{182}

3\sqrt{2x−3}−20=7

Answer

42​​​​​​​

Exercise \PageIndex{183}

\sqrt{3u−2}=\sqrt{5u+1}

​​​​​​​In the following exercise, solve.

Exercise \PageIndex{184}

A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula t=\frac{\sqrt{h}}{4} to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.

Answer

6.1 seconds


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