Solution Manual for Precalculus, 6th Edition

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INSTRUCTOR’S SOLUTIONS
MANUAL
DANIEL S. MILLER
Niagara County Community College
PRECALCULUS
SIXTH EDITION
Robert Blitzer
Miami Dade College

iii
TABLE OF CONTENTS
PRECALCULUS 6 E
Chapter P Prerequisites: Fundamental Concepts of Algebra 1....................................................1
Chapter 1 Functions and Graphs .............................................................................................127
Chapter 2 Polynomial and Rational Functions........................................................................259
Chapter 3 Exponential and Logarithmic Functions.................................................................411
Chapter 4 Trigonometric Functions ........................................................................................489
Chapter 5 Analytic Trigonometry ...........................................................................................657
Chapter 6 Additional Topics in Trigonometry ........................................................................771
Chapter 7 Systems of Equations and Inequalities ...................................................................919
Chapter 8 Matrices and Determinants ...................................................................................1057
Chapter 9 Conic Sections and Analytic Geometry................................................................1161
Chapter 10 Sequences, Induction, and Probability..................................................................1281
Chapter 11 Introduction to Calculus .......................................................................................1381

Chapter P
Prerequisites: Fundamental Concepts of Algebra
1
Section P.1
Check Point Exercises
1. 2 2
8 6( 3) 8 6(13 3)x+ − = + −
2
8 6(10)
8 6(100)
8 600
608
= +
= +
= +
=
2. a. Since 2014 is 14 years after 2000, substitute 14
for x.
2
2
4 330 3310
4(14) 330(14) 3310
8714
T x x= + +
= + +
=
The average cost of tuition and fees at public
U.S. colleges for the school year ending in
2014 was $8714.
b. The formula underestimates the actual answer
by $179.
3. The elements common to {3, 4, 5, 6, 7} and
{3, 7, 8, 9} are 3 and 7.
{3, 4,5, 6, 7} {3, 7,8,9} {3, 7}∩ =
4. The union is the set containing all the elements of
either set.
{3, 4,5, 6, 7} {3, 7,8,9} {3, 4,5, 6, 7,8,9}∪ =
5. 9, 1.3, 0, 0.3, , 9, 10
2
π 
− − 
 
a. Natural numbers: 9 because 9 3=
b. Whole numbers: 0, 9
c. Integers: 9, 0, 9−
d. Rational numbers: 9, 1.3, 0, 0.3, 9− −
e. Irrational numbers: , 10
2
π
f. Real numbers: 9, 1.3, 0, 0.3, , 9, 10
2
π
− −
6. a. 1 2−
Because 2 1.4,≈ the number inside the
absolute value bars is negative. The absolute
value of x when x < 0 is –x. Thus,
( )1 2 1 2 2 1− = − − = −
b. 3
π −
Because 3.14,
π ≈ the number inside the
absolute value bars is positive. The absolute
value of a positive number is the number itself.
Thus,
3 3.
π π− = −
c. x
x
Because 0, .x x x> =
Thus, 1
x x
x x
= =
7. 4 (5) 9 9− − = − =
The distance between –4 and 5 is 9.
8. 2 2
7(4 3 ) 2(5 )x x x x+ + +
2 2
2 2
2
7(4 3 ) 2(5 )
28 21 10 2
38 23
x x x x
x x x x
x x
= + + +
= + + +
= +
9. 6 4[7 ( 2)]x+ − −
6 4[7 2)]
6 4[9 ]
6 36 4
42 4
x
x
x
x
= + − +
= + −
= + −
= −
Concept and Vocabulary Check P.1
1. expression
2. b to the nth power; base; exponent
3. formula; modeling; models
4. intersection; A B∩
5. union; A B∪
6. natural

Chapter P Prerequisites: Fundamental Concepts of Algebra
2
7. whole
8. integers
9. rational
10. irrational
11. rational; irrational
12. absolute value; x, x
13. b a+ ; ba
14. ( )a b c+ + ; ( )ab c
15. ab ac+
16. 0; inverse; 0; identity
17. inverse; 1; identity
18. simplified
19. a
Exercise Set P.1
1. 7 5(10) 7 50 57+ = + =
2. ( )8 6 5 8 30 38+ = + =
3. 6(3) 8 18 8 10− = − =
4. ( )8 3 4 24 4 20− = − =
5. 2
8 3(8) 64 24 88+ = + =
6. ( )2
6 5 6 36 30 66+ = + =
7. 2
7 6(7) 3 49 42 3 7 3 10− + = − + = + =
8. ( )2
8 7 8 4 64 56 4 8 4 12− + = − + = + =
9. 3 3
4 5(9 7) 4 5(2)
4 5(8) 4 40 44
+ − = +
= + = + =
10. ( ) ( )
( )
3 3
6 5 8 6 6 5 2
6 5 8
6 40 46
+ − = +
= +
= + =
11. 2
8 3(8 2) 64 3(6)
64 18 46
− − = −
= − =
12. ( ) ( )2
8 4 8 3 64 4 5 64 20 44− − = − = − =
13. 5( 2) 5(10 2)
2 14 2(10) 14
5(12)
6
5 2
10
x
x
+ +
=
− −
=
= ⋅
=
14. 7( 3) 7(9 3) 7(6) 7 3 21
2 16 2(9) 16 2
x
x
− −
= = = ⋅ =
− −
15. 2 3 ; 2, 4
1
x y x y
x
+ = − =
+
( ) ( )2 2 3 4 4 12 8 8
2 1 1 1
− + − +
= = = = −
− + − −
16. 2 ; 2 and 4
2
x y x y
xy x
+ = − =
−
( )
( )( ) ( )
2 2 4 4 4 0 0
2 4 2 2 8 4 4
− + − +
= = =
− − − − +
17. 5 (50 32)
9
C = − 5 (18) 10
9
= =
50°F is equivalent to 10°C.
18. 5 5 5
( 32) (86 32) (54) 30
9 9 9
C F= − = − = =
86°F is equivalent to 30°C.
19. 2 2
4 60 16 4 60(2) 16(2)
4 120 16(4) 4 120 64
124 64 60
h t t= + − = + −
= + − = + −
= − =
Two seconds after it is kicked, the ball’s height is
60 feet.
20. 2
2
4 60 16
4 60(3) 16(3)
4 180 16(9)
4 180 144
184 144 40
h t t= + −
= + −
= + −
= + −
= − =
Three seconds after it is kicked, the ball’s height is
40 feet.
21. { } { } { }1, 2,3, 4 2, 4,5 2, 4=
22. { } { } { }1,3, 7 2,3,8 3=
23. { } { } { }, , , , , ,s e t t e s s e t=

Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers
3
24. { } { } { }, , , , , , , , ,r e a l l e a r r e a l=
25. { } { } { }1,3,5, 7 2, 4, 6,8,10 =
The empty set is also denoted by .∅
26. { } { } { }1,3,5, 7 5, 3, 1− − − = or ∅
27. { }, , ,a b c d ∅ = ∅
28. { }, ,w y z ∅ = ∅
29. { } { } { }1, 2,3, 4 2, 4,5 1, 2,3, 4,5=
30. { } { } { }1,3, 7,8 2,3,8 1, 2,3, 7,8=
31. { } { }
{ }
1,3,5, 7 2, 4, 6,8,10
1, 2,3, 4,5, 6, 7,8,10=

32. { } { } { }0,1,3,5 2, 4, 6 0,1, 2,3, 4,5, 6=
33. { } { }, , , , , , , ,a e i o u a e i o u∅ =
34. { } { }, , , , , , , ,e m p t y e m p t y∅ =
35. a. 100
b. 0, 100
c. 9, 0, 100−
d. 4
9, , 0, 0.25, 9.2, 100
5
− −
e. 3
f. 4
9, , 0, 0.25, 3,9.2, 100
5
− −
36. a. 49
b. 0, 49
c. 7, 0, 49−
d. 7, 0.6, 0, 49− −
e. 50
f. 7, 0.6, 0, 49, 50− −
37. a. 64
b. 0, 64
c. 11, 0, 64−
d. 5
11, , 0, 0.75, 64
6
− −
e. 5, π
f. 5
11, , 0, 0.75, 5, , 64
6
π− −
38. a. 4
b. 0, 4
c. 5, 0, 4−
d. 5, 0.3, 0, 4− −
e. 2
f. 5, 0.3, 0, 2, 4− −
39. 0
40. Answers will vary. An example is 1 .
2
41. Answers will vary. An example is 2.
42. Answers will vary. An example is 2.−
43. true; –13 is to the left of –2 on the number line.
44. false; –6 is to the left of 2 on the number line.
45. true; 4 is to the right of –7 on the number line.
46. true; –13 is to the left of –5 on the number line.
47. true; −π = −π
48. true; –3 is to the right of –13 on the number line.
49. true; 0 is to the right of –6 on the number line.
50. true; 0 is to the right of –13 on the number line.
51. 300 300=
52. 203 203− =

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Chapter P Prerequisites: Fundamental Concepts of Algebra
4
53. 12 12− π = − π
54. 7 7− π = − π
55. 2 5 5 2− = −
56. 5 13 13 5− = −
57. 3 3 1
3 3
− −
= = −
−
58. 7 7 1
7 7
− −
= = −
−
59. 3 7 3 7 4 4− − − = − = − =
60. 5 13 5 13 8 8− − − = − = − =
61. 2 ( 5) 3 3x y+ = + − = − =
62. 2 ( 5) 7 7x y− = − − = =
63. 2 5 2 5 7x y+ = + − = + =
64. 2 5 2 5 3x y− = − − = − = −
65. 5 5 1
5 5
y
y
− −
= = = −
−
66. 2 5 2 5 1 ( 1) 0
2 5 2 5
x y
x y
−
+ = + = + = + − =
− −
67. The distance is 2 17 15 15− = − = .
68. The distance is 4 15 11 11− = − = .
69. The distance is 2 5 7 7− − = − = .
70. The distance is 6 8 14 14− − = − = .
71. The distance is 19 ( 4) 19 4 15 15− − − = − + = − = .
72. The distance is 26 ( 3) 26 3 23 23− − − = − + = − = .
73. The distance is
3.6 ( 1.4) 3.6 1.4 2.2 2.2− − − = − + = − = .
74. The distance is
5.4 ( 1.2) 5.4 1.2 4.2 4.2− − − = − + = − = .
75. 6 + (–4) = (–4) + 6;
commutative property of addition
76. 11 (7 4) 11 7 11 4;⋅ + = ⋅ + ⋅
distributive property of multiplication over addition
77. 6 + (2 + 7) = (6 + 2) + 7;
associative property of addition
78. 6 (2 3) 6 (3 2);⋅ ⋅ = ⋅ ⋅
commutative property of multiplication
79. (2 + 3) + (4 + 5) = (4 + 5) + (2 + 3);
commutative property of addition
80. 7 (11 8) (11 8) 7;⋅ ⋅ = ⋅ ⋅
commutative property of multiplication
81. 2 (–8 + 6) = –16 + 12;
distributive property of multiplication over addition
82. 8(3 11) 24 ( 88)− + = − + − ;
distributive property of multiplication over addition
83. ( )
1 3 1; 3
3 x x
x + = ≠ −
+ ,
inverse property of multiplication
84. ( ) ( )4 4 0;x x+ + − + =  
inverse property of addition
85. 5(3 4) 4 5 3 5 4 4
15 20 4
15 16
x x
x
x
+ − = ⋅ + ⋅ −
= + −
= +
86. 2(5 4) 3 2 5 2 4 3
10 8 3
10 5
x x
x
x
+ − = ⋅ + ⋅ −
= + −
= +
87. 5(3 2) 12 5 3 5 2 12
15 10 12
27 10
x x x x
x x
x
− + = ⋅ − ⋅ +
= − +
= −
88. 2(5 1) 14 2 5 2 1 14
10 2 14
24 2
x x x x
x x
x
− + = ⋅ − ⋅ +
= − +
= −
89. 7(3 5) 2(4 3)
7 3 7 5 2 4 2 3
21 35 8 6
29 29
y y
y y
y y
y
− + +
= ⋅ − ⋅ + ⋅ + ⋅
= − + +
= −

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Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers
5
90. 4(2 6) 3(5 10)
4 2 4 6 3 5 3 10
8 24 15 30
23 6
y y
y y
y y
y
− + +
= ⋅ − ⋅ + ⋅ + ⋅
= − + +
= +
91. ( ) ( )5 3 2 7 2 15 10 7 2
8 12
y y y y
y
− − + = − − −
= −
92. 4(5 3) (6 3) 20 12 6 3
14 15
y y y y
y
− − + = − − −
= −
93. ( ) [ ]7 4 3 4 5 7 4 3 4 5y y− − − = − − +  
[ ]7 4 8 4
7 32 16
16 25
y
y
y
= − −
= − +
= −
94. [ ]
[ ]
6 5[8 (2 4)] 6 5 8 2 4
6 5 12 2
6 60 10
10 54
y y
y
y
y
− − − = − − +
= − −
= − +
= −
95. ( )
( )
2 2
2 2
2 2
2 2
2 2
2 2
18 4 6 2 5
18 4 6 12 5
18 4 6 7
18 4 6 7
18 6 4 7
18 6 11 12 11
x x
x x
x x
x x
x x
x x
 + − − +
 
 = + − − +
 
 = + − −
 
= + − +
= − + +
= − + = +
96. ( )
( )
2 2
2 2
2 2
2 2
2 2
2
2
14 5 7 2 4
14 5 7 14 4
14 5 7 10
14 5 7 10
14 7 5 10
14 7 15
7 15
x x
x x
x x
x x
x x
x
x
 + − − +  
 = + − − +
 
 = + − −
 
= + − +
= − + +
= − +
= +
97. –(–14x) = 14x
98. ( )17 17y y− − =
99. –(2x – 3y – 6) = –2x + 3y + 6
100. ( )5 13 1 5 13 1x y x y− − − = − + +
101. [ ]
1 (3 ) (4 ) ( 4 ) 0
3 x y y x x+ + − = + =
102. ( ) ( )
1 2 7 7 0
2 y x x y y+ − + = + =  
103. 6 3
6 3
6 3
− −
>
Since 6 3, 6 3 .> − > −
104. 20 50
20 50
20 50
− −
<
Since 20 50, 20 50 .< − < −
105. 3 0.6
5
0.6 0.6
0.6 0.6
0.6 0.6
−
−
=
Since 3
0.6 0.6, 0.6 .
5
= = −
106. 5 2.5
2
2.5 2.5
.5 2.5
2.5 2.5
2
−
−
=
Since 5
2.5 2.5,

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Chapter P Prerequisites: Fundamental Concepts of Algebra
6
110. 4 4
2 17 17
4 17
2 17 4
2 1
2 1
− ÷
⋅
>
Since 4 4
2 1, 2 > .
17 17
> − ÷
111. 2 2
8 16 2 4 3 64 16 4 4 3
64 4 4 3
64 16 3
48 3
45
− ÷ ⋅ − = − ÷ ⋅ −
= − ⋅ −
= − −
= −
=
112. 2 2
10 100 5 2 3 100 100 25 2 3
100 4 2 3
100 8 3
92 3
89
− ÷ ⋅ − = − ÷ ⋅ −
= − ⋅ −
= − −
= −
=
113.
2
2 2 2
2
2
5 2 3 5 2 9
[3 ( 2)] [9 ( 2)]
10 9
[9 2]
10 9
11
1
121
⋅ − ⋅ −
=
− − − −
−
= +
−
=
=
114. 2 2
2
10 2 3 4 5 12
(12 3 2) (12 6)
17
6
17
36
÷ + ⋅ +
=
− ⋅ −
=
=
115. 8 3[ 2(2 5) 4(8 6)] 8 3[ 2( 3) 4(2)]
8 3[6 8]
8 3[ 2]
8 6
14
− − − − − = − − − −
= − −
= − −
= +
=
116. 8 3[ 2(5 7) 5(4 2)] 8 3[ 2( 2) 5(2)]
8 3[4 10]
8 3[ 6]
8 18
26
− − − − − = − − − −
= − −
= − −
= +
=
117. 2( 2) 4( 3) 4 12
5 8 3
8
3
8
3
− − − − +
=
− −
= −
= −
118. 6( 4) 5( 3) 24 15
9 10 1
9
1
9
− − − − +
=
− −
−
= −
=
119.
2 2
2
(5 6) 2 3 7 ( 1) 2 4
89 3 2589 3 5 1 2(4)
89 75
1 8
14
7
14
1
2
− − − − − −
= − ⋅− ⋅ −
= −
−
=
−
=
= −
120.
2 2
2
12 3 5 2 3 12 3 5 4 9
7 3 367 3 6 4 5 13
10 36
20(13)
26
260
26
10
÷ ⋅ + ÷ ⋅ +
= + −+ − ⋅
= −
= −
= −
= −
121. ( )4 4 4x x x x− + = − − = −
122. ( )8 8 2 8x x x x x− − = − + = −
123. ( )6 5 30x x− = −
124. ( )10 4 40x x− = −
125. 5 2 3x x x− =
126. ( )6 2 6 2 8x x x x x− − = + =
127. ( )8 3 6 8 3 6 5 6x x x x x− + = − − = −
128. ( )8 3 6 8 3 18 3 10x x x− + = − − = − −
129. a. ( )
( )
( )
7 220
10
7 220 20
10
7 200
10
140
H a
H
= −
= −
=
=
The lower limit of the heart rate for a 20-year-
old with this exercise goal is 140 beats per
minute.

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Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers
7
b. ( )
( )
( )
4 220
5
4 220 20
5
4 200
5
160
H a
H
= −
= −
=
=
The upper limit of the heart rate for a 20-year-
old with this exercise goal is 160 beats per
minute.
130. a. ( )
( )
( )
1 220
2
1 220 30
2
1 190
2
95
H a
H
= −
= −
=
=
The lower limit of the heart rate for a 30-year-
old with this exercise goal is 95 beats per
minute.
b. ( )
( )
( )
3 220
5
3 220 30
5
3 190
5
114
H a
H
= −
= −
=
=
The upper limit of the heart rate for a 30-year-
old with this exercise goal is 114 beats per
minute.
131. a. 2
2
21 862 15,552
21(14) 862(14) 15,552
31, 736
T x x= + +
= + +
=
The formula estimates the cost to have been
$31,736 in 2014.
b. This overestimates the value in the graph by
$35.
c. 2
2
21 862 15,552
21(20) 862(20) 15,552
41,192
T x x= + +
= + +
=
The formula projects the cost to be $41,192 in
2020.
132. a. 2
2
21 862 15,552
21(12) 862(12) 15,552
28,920
T x x= + +
= + +
=
The formula estimates the cost to have been
$28,920 in 2012.
b. This underestimates the value in the graph by
$136.
c. 2
2
21 862 15,552
21(22) 862(22) 15,552
44, 680
T x x= + +
= + +
=
The formula projects the cost to be $44,680 in
2022.
133. a. ( )0.05 0.12 10, 000
0.05 1200 0.12
1200 0.07
x x
x x
x
+ −
= + −
= −
b. 1200 0.07 1200 0.07(6000)
$780
x− = −
=
134. a. 0.06 0.5(50 ) 0.06 25 0.5
25 0.44
t t t t
t
+ − = + −
= −
b. 0.06(20) 0.5(50 20)
1.2 0.5(30)
1.2 15
16.2 miles
+ −
= +
= +
=
135. – 143. Answers will vary.
144. does not make sense; Explanations will vary.
Sample explanation: Models do not always
accurately predict future values.
145. does not make sense; Explanations will vary.
Sample explanation: To use the model, substitute 0
for x.
146. makes sense
147. does not make sense; Explanations will vary.
Sample explanation: The commutative property
changes order and the associative property changes
groupings.
148. false; Changes to make the statement true will vary.
A sample change is: Some rational numbers are not
integers.
149. false; Changes to make the statement true will vary.
A sample change is: All whole numbers are integers.
150. true
151. false; Changes to make the statement true will vary.
A sample change is: Some irrational numbers are
negative.
152. false; Changes to make the statement true will vary.
A sample change is: The term x has a coefficient
of 1.

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Chapter P Prerequisites: Fundamental Concepts of Algebra
8
153. false; Changes to make the statement true will vary.
A sample change is:
5 3( 4) 5 3 12 3 7.x x x+ − = + − = −
154. false; Changes to make the statement true will vary.
A sample change is: 2 .x x x− − = −
155. true
156. 2 1.4
1.4 1.5
2 1.5
≈
<
<
157. 3.5
π− > −
158. 3.14 1.57
2
1.571
2
1.57 1.571
3.14
2 2
− = −
π
− ≈ −
− > −
π
− > −
159. a. 4 3 7
( )( )b b b b b b b b b b⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ =
b. 5 5 10
( )( )b b b b b b b b b b b b b⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =
c. add the exponents
160. a.
7 4
3
b b b b b b b b b
b b bb
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= =
⋅ ⋅
b.
8 6
2
b b b b b b b b b b
b bb
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= =
⋅
c. subtract the exponents
161. 3
6.2 10 6.2 10 10 10 6200× = × × × =
It moves the decimal point 3 places to the right.
Section P.2
Check Point Exercises
1. a. ( ) ( ) ( ) ( )
4 4 443 6 3 6 12 24
2 2 16x y x y x y= =
b. ( )( ) ( )2 5 3 2 5 3
3 8
6 3 6 3
18
x y xy x x y y
x y
− = − ⋅ ⋅ ⋅ ⋅ ⋅
= −
c.
12 2 12 2
16 4 16 4
100 100
2020
x y x y
x y x y− −
   
=           
12 16 2 ( 4)
4 6
6
4
5
5
5
x y
x y
y
x
− − −
−
=
=
=
d. ( ) ( )
( )
2 2 2
4 24
55 xx
y y
− − −
−
  =  
 
( ) ( )
( )
2 2
24
2 2
8
8
2 2
8
2
5
5
5
25
x
y
x
y
y
x
y
x
− −
−
− −
−
=
=
=
=
2. a. 9
2.6 10 2, 600, 000, 000− × = −
b. 6
3.017 10 0.000003017−
× =
3. a. 9
5, 210, 000, 000 5.21 10= ×
b. 8
0.00000006893 6.893 10−
− = − ×
4. ( )7 2 7
410 10 4.1 10 10× = × ×
( )2 7
9
4.1 10 10
4.1 10
= × ×
= ×
5 a. ( )( )5 7
7.1 10 5 10−
× ×
( )
( )
5 7
2
1 2
1 2
1
7.1 5 10 10
35.5 10
3.55 10 10
3.55 10 10
3.55 10
−
−
−
−
−
= ⋅ × ⋅
= ×
= × ×
= × ×
= ×
b.
6 6
3 3
1.2 10 1.2 10
33 10 10− −
× = ⋅
×
6 ( 3)
9
8
0.4 10
0.4 10
4 10
− −
= ×
= ×
= ×

Loading page 11...

Section P.2 Exponents and Scientific Notation
9
6.
10 10 10
5 5
5
4.08 10 4.08 10 4.08 10
680, 000 6.86.8 10 10
0.6 10
60, 000
× ×
= = ⋅
×
= ×
=
The average salary was $60,000 per U.S. police
officer.
7. 5 –2 2
5 –2 2
3
(1.76 10 )[(1.44 10 ) ]
(1.76 10 )[(1.44 10 ) 0 ]
2.5344 10
2534.4
S r= × × −
= × × −
= ×
=
The speed of the blood at the central axis of the
artery is 2534.4 centimeters per second.
Concept and Vocabulary Check P.2
1. m n
b + ; add
2. m n
b − ; subtract
3. 1
4. 1
n
b
5. false
6. n
b
7. true
8. a number greater than or equal to 1 and less than 10;
integer
9. true
10. false
Exercise Set P.2
1. 2
5 2 (5 5) 2 25 2 50⋅ = ⋅ ⋅ = ⋅ =
2. 2
6 2 (6 6) 2 36 2 72⋅ = ⋅ ⋅ = ⋅ =
3. 6
( 2) ( 2)( 2)( 2)( 2)( 2)( 2) 64− = − − − − − − =
4. 4
( 2) ( 2)( 2)( 2)( 2) 16− = − − − − =
5. 6
2 2 2 2 2 2 2 64− = − ⋅ ⋅ ⋅ ⋅ ⋅ = −
6. 4
2 2 2 2 2 16− = − ⋅ ⋅ ⋅ = −
7. 0
( 3) 1− =
8. 0
( 9) 1− =
9. 0
3 1− = −
10. 0
9 1− = −
11. 3
3
1 1 1
4 4 4 4 644
− = = =
⋅ ⋅
12. 6
6
1 1 1
2 2 2 2 2 2 2 642
− = = =
⋅ ⋅ ⋅ ⋅ ⋅
13. 2 3 2 3 5
2 2 2 2 2 2 2 2 2 32+
⋅ = = = ⋅ ⋅ ⋅ ⋅ =
14. 3 2 3 2 5
3 3 3 3 3 3 3 3 3 243+
⋅ = = = ⋅ ⋅ ⋅ ⋅ =
15. 2 3 2 3 6
(2 ) 2 2 2 2 2 2 2 2 64⋅
= = = ⋅ ⋅ ⋅ ⋅ ⋅ =
16. 3 2 3 2 6
(3 ) 3 3 3 3 3 3 3 3 729⋅
= = = ⋅ ⋅ ⋅ ⋅ ⋅ =
17.
8 8 4 4
4
2 2 2 2 2 2 2 16
2
−
= = = ⋅ ⋅ ⋅ =
18.
8 8 4 4
4
3 3 3 3 3 3 3 81
3
−
= = = ⋅ ⋅ ⋅ =
19. 3 3 1 2
2
1 1 1
3 3 3 3 3 3 93
− − + −
⋅ = = = = =
⋅
20. 3 3 1 2
2
1 1 1
2 2 2 2 2 2 42
− − + −
⋅ = = = = =
⋅
21.
3 3 7 4
7 4
2 1 1 1
2 2 2 2 2 2 162 2
− −
= = = = =
⋅ ⋅ ⋅
22.
4 4 7 3
7 3
3 1 1 1
3 3 3 3 3 273 3
− −
= = = = =
⋅ ⋅
23. 2
2 2
1 y
x y y
x x
− = ⋅ =
24. 3
3 3
1 x
xy x y y
− = ⋅ =
25. 0 5 5 5
1x y y y= ⋅ =
26. 7 0 7 7
1x y x x⋅ = ⋅ =

Loading page 12...

Chapter P Prerequisites: Fundamental Concepts of Algebra
10
27. 3 7 3 7 10
x x x x+
⋅ = =
28. 11 5 11 5 16
x x x x+
⋅ = =
29. 5 10 5 10 5
x x x x− − +
⋅ = =
30. 6 12 6 12 6
x x x x− − +
⋅ = =
31. 3 7 3 7 21
( )x x x⋅
= =
32. 11 5 11 5 55
( )x x x⋅
= =
33. 5 3 5 3 15
15
1
( )x x x x
− − ⋅ −
= = =
34. 6 4 6 4 24
24
1
( )x x x x
− − ⋅ −
= = =
35.
14 14 7 7
7
x x x
x
−
= =
36.
30 30 10 20
10
x x x
x
−
= =
37.
14 14 ( 7) 14 7 21
7
x x x x
x
− − +
− = = =
38.
30 30 ( 10) 30 10 40
10
x x x x
x
− − +
− = = =
39. 3 2 2 3 2 2 3 2 6
(8 ) 8 ( ) 8 64x x x x⋅
= = =
40. 4 2 2 4 2 2 4 2 8
(6 ) (6) ( ) 6 36x x x x⋅
= = =
41.
3 3
3 3
4 ( 4) 64
x x x
− 
− = = − 
 
42.
3 3
3 3
6 ( 6) 216
y y y
  −
− = = − 
 
43. 2 5 2 2 2 2 5 2
2 2 5 2
4 10
( 3 ) ( 3) ( ) ( )
9
9
x y x y
x y
x y
⋅ ⋅
− = − ⋅
=
=
44. 4 6 3 3 4 3 6 3
4 3 6 3
12 18
( 3 ) ( 3) ( ) ( )
27
27
x y x y
x y
x y
⋅ ⋅
− = −
= −
= −
45. 4 7 4 7 4 7 11
(3 )(2 ) 3 2 6 6x x x x x

Loading page 13...

Section P.2 Exponents and Scientific Notation
11
56. 2 3 3 2 ( 3)
3 6
3 6
6
(10 ) 10
10
1
10 1
1000
x x
x
x
x
− − ⋅ −
− −
=
=
=
=
57.
3 5 3 7 5 ( 9)
7 9
4 14
14
4
24 3
432 3
4
3
4
x y x y
x y
x y
y
x
− − −
−
−
⋅ =
=
=
58.
4 9 4 12 9 ( 3)
12 3
8 12
12
8
10 1
330 1
3
3
x y x y
x y
x y
y
x
− − −
−
−
=
=
=
59.
23 2 6 2
6
5 5
2 25
x x y
y xy
− − −  = = 
  −
 
60.
3 34
4
3
3 4 3
3
12
3
3
3
27
x y
y x
y
x
y
x
−
⋅
   
=       
=
=
61.
( ) 33 2 34 2
10 3 10 4
35
6
15
18
15 3
5
3
27
a b b
a b a
b
a
b
a
− −
− −
  − −
 = 
      
 −
=  
 
 
−
=
62.
( ) 33 8 214 8
17 2 17 14
310
30
9
30 3
10
3
3
27
a b b
a b a
b
a
b
a
− −
− −
  − −
 = 
      
 −
=  
 
 
−
=
63.
05 2
3 4
3 1
12
a b
a b
−
−
  = 
 
 
64.
05 3
3 5
4 1
12
a b
a b
−
−
  = 
 
 
65. 2
3.8 10 380× =
66. 2
9.2 10 920× =
67. 4
6 10 0.0006−
× =
68. 5
7 10 0.00007−
× =
69. 6
7.16 10 7,160, 000− × = −
70. 6
8.17 10 8,170, 000− × = −
71. 1
7.9 10 0.79−
× =
72. 1
6.8 10 0.68−
× =
73. 3
4.15 10 0.00415−
− × = −
74. 3
3.14 10 0.00314−
− × = −
75. 10
6.00001 10 60, 000,100, 000− × = −
76. 10
7.00001 10 70, 000,100, 000− × = −
77. 4
32, 000 3.2 10= ×
78. 4
64, 000 6.4 10= ×
79. 17
638, 000, 000, 000, 000, 000
6.38 10= ×
80. 17
579, 000, 000, 000, 000, 000 5.79 10= ×
81. 3
5716 5.716 10− = − ×
82. 3
3829 3.829 10− = − ×
83. 3
0.0027 2.7 10−
= ×
84. 3
0.0083 8.3 10−
= ×

Loading page 14...

Chapter P Prerequisites: Fundamental Concepts of Algebra
12
85. 9
0.00000000504 5.04 10−
− = − ×
86. 9
0.00000000405 4.05 10−
− = − ×
87. ( )( ) ( )( )4 3 4 3
4 3 7
3 10 2.1 10 3 2.1 10 10
6.3 10 6.3 10+
× × = × ×
= × = ×
88. ( )( )4 3 7
2 10 4.1 10 8.2 10× × = ×
89. ( )( ) ( )( )15 11 15 11
15 ( 11)
4
1.6 10 4 10 1.6 4 10 10
6.4 10
6.4 10
− −
+ −
× × = × ×
= ×
= ×
90. ( )( ) ( )( )15 11 15 11
15 ( 11)
4
1.4 10 3 10 1.4 3 10 10
4.2 10
4.2 10
− −
+ −
× × = × ×
= ×
= ×
91. ( )( ) ( )( )8 4 8 4
8 ( 4)
12
11
6.1 10 2 10 6.1 2 10 10
12.2 10
12.2 10
1.22 10
− − − −
− + −
−
−
× × = × ×
= ×
= ×
= ×
92. ( )( )8 4 12
11
5.1 10 3 10 15.3 10
1.53 10
− − −
−
× × = ×
= ×
93. ( )( )
( )( )
8 4
8 4
8 4
12
13 13
4.3 10 6.2 10
4.3 6.2 10 10
26.66 10
26.66 10
2.666 10 2.67 10
+
× ×
= × ×
= ×
= ×
= × ≈ ×
94. ( )( )8 4
8 4 12
13 13
8.2 10 4.6 10
37.72 10 37.72 10
3.772 10 3.77 10
+
× ×
= × = ×
= × ≈ ×
95.
8 8
5 5
8 5 3
8.4 10 8.4 10
44 10 10
2.1 10 2.1 10−
× = ×
×
= × = ×
96.
8 8 5 3
5
6.9 10 2.3 10 2.3 10
3 10
−× = × = ×
×
97. 4 4
2 2
4 ( 2)
6 5
3.6 10 3.6 10
99 10 10
0.4 10
0.4 10 4 10
− −
− −
× = ×
×
= ×
= × = ×
98.
( )
4 4 ( 2) 6
2
1 6 5
1.2 10 0.6 10 0.6 10
2 10 6 10 10 6 10
− −
−
−
× = × = ×
× = × × = ×
99.
2 2
6 6
2 6 8
4.8 10 4.8 10
2.42.4 10 10
2 10 2 10
− −
− − −
× = ×
×
= × = ×
100.
2 2 6 8
6
7.5 10 3 10 3 10
2.5 10
− − − −× = × = ×
×
101. 2 2
6 6
2 ( 6)
4 3
2.4 10 2.4 10
4.84.8 10 10
0.5 10
0.5 10 5 10
− −
− −
− − −
× = ×
×
= ×
= × = ×
102. 2 2 ( 6)
6
4 3
1.5 10 0.5 10
5 10
0.5 10 5 10
− − − −
−
× = ×

Loading page 15...

Section P.2 Exponents and Scientific Notation
13
106. ( )( )
( )( )( )
4 3
3 3
1 1 6
6
7
6.6 10 1 1066000 0.001
0.003 0.002 3 10 2 10
6.6 10 1.1 10
6 10
1.1 10
−
− −
− −
−
× ×× =
× × ×
×
= = ×
×
= ×
107. ( )
( ) ( )
32 6 3
3 6 32 1
3 36 6 0 0 1
x y x y
x yx y
x y x y
−− −
−−
− − −−
=
= = =
108. ( )
( ) ( )
22 2 4
3 6 32
7
4 32 6 8 7
8
xy x y
x yx y
y
x y x y x
−− −
− −−
− −− − −
=
= = =
109. ( )( ) 53 6 3 6 5 5
4 8 6
4 8 6 8 6
2 2 2 2
2 2 16
x yz x x yz x
y y
x yz x z x z
−− − − − − −
− − −
= ⋅ ⋅
= = =
110. ( )( ) 34 7 4 7 3 3
2 7 7
2 7 7 7 7
3 3 3 3
3 3 9
x yz x x yz x
y y
x yz x z x z
−− − − − − −
− − −
= ⋅ ⋅
= = =
111. ( )
23 4 5 26 8 10
3 4 5
12 16 20
12 16 20
1
x y z x y z
x y z
x y z x y z
− −
− − −
− − −
  =  
 
= =
112. ( )
44 5 6 48 10 12
4 5 6
32 40 48
32 40 48
1
x y z x y z
x y z
x y z x y z
− −
− − −
− − −
  = 
 
 
= =
113. ( ) ( ) ( )
( )
( )( ) ( )
( )
2 2 01 2 1 4 3 3 3
23 5
2 2 2 2 8 6
2 6 10
18 6
2 2 16
2
2 2 1
2
4
x y x y x y
x y
x y x y
x y
x y
− −− − − − −
− −
− −
− −
=
=
114. ( ) ( ) ( )
( )
( )( )( )
( )
2 2 01 3 1 6 4 3 3
24 6
2 6 2 2 12 8
2 8 12
26 6
2 2 9
2
2 2 1
2
4
x y x y x y
x y
x y x y
x y
x y
− −− − − − −
− −
− −
− −
=
=
115. a. 12
3.18 10×
b. 8
3.20 10×
c.
12 12
8 8
4
3.18 10 3.18 10
3.203.20 10 10
0.9938 10
9938
× = ×
×
≈ ×
≈
$9938 per American
116. a. 12
3.02

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