Review Sheet – Math 096 – Final Exam
Simplify. If the expression does not represent a real number, say so. Assume all variables
represent positive real numbers.
1.
81
2.
5.
50
6.
€
€
240
7.
3
10. 64
4.
3
125x 6 y 12
48
8.
5
a 7 b5c 23
12
€
€
⎛100 ⎞ 3 2
18. ⎜
⎟
⎝169 ⎠
€
€
12. ( −49)
12
€
14. 625−1 4
⎛ 1 ⎞ −5 3
17. ⎜ ⎟
⎝ 8 ⎠
−32
11. −36
€
23
€
!
5
12
45x y z
13. ( −27)
€
3.
€
8 13 19
€
€
16
€
9.
€
4
15. 64 −2
3
16. −144 −1 2
€
€
€
€
Simplify, using the rules of radicals. Assume all variables represent positive real numbers.
€
9
19. 3⋅ 15
20. 2x ⋅ 2x
21.
22. 6 + 2 54 " 7 24
4a 8
23. 7 20x + 6 45x " 3 80x
€
(
)(
25. 8 + 3 2 9 − 2 2
24. 5x 3 xy 2 + 7 3 8x 4 y 2
!
€
)
(
26. 12 − 3
€
)
2
(
)(
27. 11 − 7 11+ 7
28. Find the perimeter and area of the given triangle. Simplify your result.
€
€
75 in
75 in
27 in
€
€
192 in
€
Use the rules for exponents to simplify each expression. Write the answer with positive
exponents.€ Assume that all variables represent positive real numbers.
(a )
2 3
29. x 1 4 ⋅ x 2
€
3
30.
€
4
31.
a5 3
€
(
x 2 3 y −4 5
)
15
32.
€
t −1
t −1 5 ⋅ t 7 10
)
Write each radical in exponent form.
33.
€
€
€
r4
34.
€
€
1
35.
x
3
€
b
−6
38.
39.
1+ 5
4 7
2− 3
63
y3
€
€
€
Factor each of the following polynomials completely.
41. x 2 + 25x
42. 4x 4 y 2 −12x 3 y 3
43. 15z 8 + 25z 6 + 5z 3
44. m 2 −121
45. 16 y 2 − 225t 2
46. y 3 + 6 y 2 − 9 y − 54
€
2
47. x − 7x +10
€
5
Rationalize each€denominator.
2
5
36.
37.
6
12
40.
€
7
€
50. 3k 2 + 21k +18
3
2
5
4
53. 2a + 36a b + 64ab
56. 6 y − 22 y +12 y
€
3
2
€
€
€
2
48. k − 8k − 20
51. 6x 2 +17x + 5
2
54. 10x −19x + 6
€
€
€
4
57. a −16
€
49. z 2 + 6z −16
52. 8x 2 +10x − 7
55. 8m 2 −12m − 80
58. 49x 3 − x
€
€
€
€
Find any values of the variable for which each rational expression is undefined. Write the
answers with the symbol ≠.
y +6
8x
k +1
6
59.
60. 2
61.
62.
x −7
x +9
k 2 + 9k − 22
2y2 + y
€
€
Write each rational expression in lowest€terms.
5a 3b
6a −1
m+5
63.
64.
65.
2 7
m −5
35a b
1 − 36a 2
€
3z 2 +10z + 8
67. 2
5z +13z + 6€
16x 2 + 24x
68.
32x
€
€
€
€
69.
€
x 2 − 6x − 7
x 2 + 8x + 7 €
66.
4x +12
6x +18
Perform the indicated operations. Write each answer in lowest terms.
x 2 − 25
4x 2 − 20x
z 2 + 4z − 5 5z
÷
⋅
70.
71. 2
5z − 5
3z +15
x − x − 30 x 2 + 2x − 48
€
72.
75.
€
78.
€
€
€
2t 2 + 7t + 6
4 −t
2
4
5b
+
b b+2
÷
2t 2 − 5t −12
73.
2
t + 5t −14
76.
€
x
2
+ 2
x − 25 €x + 8x +15
€
€
x +4
−
x 2 + 6x + 5 €x 2 + 5x + 4
−6
2
x +10x + 24
−
3
x +6
x +4
€
Solve each quadratic equation. If a method is specified, use that method. Otherwise, use the
method of your choice: factoring, square root property, completing the square, or quadratic
€
formula.
81. Factor: x 2 − 3x − 28 = 0
82. Factor: 10x 2 + 25x − 60 = 0
2
83. Square Root Property: 2m 2 −10 = 4
€
84. Square Root Property: (5x −1) = 28
€
85. Complete the Square: x 2 + 6x = 5
€
86. Complete the Square: y 2 − 8 y − 2 = 0
€
88. Quadratic Formula: 5z 2 = 8z − 2
87. Quadratic Formula: 2x + x − 8 = 0
€
2
89. x = 3x
90. 8 − 4t = 3t 2
€
92. (5x +1)( x − 2) = 6
93. ( x − 8)( x + 2) = 24
€
€
2
2
95. 9k − 9k = 27
96. 9b − 36 = 0
€
€
91. ( x − 7)( x − 3) = 10
€
94. 4x ( x + 2) = 7
€
Solve each rational equation. Be sure to check your solutions.
€
−3
x
x2 + 2
x
3x
2
1
1
+
= 2
97. x + = 9 +
98. +
99.
=
x + 8 x + 6 x +14x + 48
2
5
3 x +2 x
100.
€
77.
7
3
+
y +5 y −5
Simplify each complex fraction.
4y − 8
6
2+
16
x
79.
80.
6 y −12
9
1− 2
4
x
€
€
74.
2
2
€
9
5
−
4x 6x
4
2
8
−
= 2
x +5 x −5 x €
− 25
101.
€
x
1
3
− 2
=−
x −1 x − x
x
€
Solve each radical equation. Be sure to check your solutions.
102.
2x − 5 = 3
105.
103. 3 x = 8x +16
x +1 + 5 = x
106.
€
€
x + 2 − x − 3 =1
€
10x + 24 = x + 4
104.
107.
3
2x + 5 + 4 = 7
€
Solve each formula for the specified variable.
€
€
Mn
1 a
108. r =
for F
109. N =
for a
F
2π r
110. k =
rF
wv 2
for v
111. V = πr 2 h for r
€
€
€
€ Solve each problem.
112. The height of a rectangle measures 12 inches. Its diagonal measures 15 inches. What is
the width of the rectangle?
113. A 13-foot ladder is leaning against a house. The distance from the bottom of the ladder to
the house is 7 feet less than the distance from the top of the ladder to the ground. How far is the
bottom of the ladder from the house?
114. The area of a triangle is 45 square inches. The base is 3 inches more than three times the
height. Find the measures of the base and the height of the triangle.
115. The length of a rectangle is 1 meter less than twice the width. The area of the rectangle is
153 square meters. Find the length and the width of the rectangle.
116. A right triangle has one leg that is 2 cm longer than the other. The hypotenuse is 2 cm less
than twice the length of the shorter leg. Find the length of the shorter leg.
Find the following values, given the functions.
117. g ( x ) = x 2 − 8x +11
a. g ( −10)
b. g (0)
c. g (4)
d. g ( z)
€ 118. h ( x ) = −4x + 7
€
a. h (0)
€
119. f ( x ) =
€
€
b. h ( −3)
2x +1
3x€− 4
a. f (2)
b. f ( −1)
€
€
⎛ 1 ⎞
c. h⎜ ⎟
⎝ 2 ⎠
€
€
( )
d. f a 2
c. f ( −x )
€
€
€
€
d. h ( p + t )
€
State the domain and range of the following relations. Then state whether the relation is a
function or not. Explain your reasoning.
120.
{(3,4), (5, −2), (0,6), (1,8), (−7,4), (−4, −1)}
122.
121.
{(2,5), (3,7), (4,9), (2,11)}
123.
124.
€
€
125.
126.
128. Refer to the following graph of f ( x ) to answer the questions.
a. What is f (2) ?
€
b. For what values of x does f ( x ) = 4 ?
€
€
127.
129. Graph each of the following functions by plotting points.
a. f ( x ) = −3x + 2
b. f ( x ) = −x 2
e. f ( x ) = −5
f. f ( x ) = x + 4
€
€
€
€
c. f ( x ) = x +1
€
1. 9
2. 2
3. −2
4. 5x 2 y 4
5. 5 2
6. 4 15
7. 2 3 6
8. abc 4
€
9. 3x y z
€
5yz
€
13. 9
17. 32
18.
€
19. 3 5
€
€
€
€
€
€
1
15.
16
16. −
21.
3
22. −7 6
2a 4
25. 60 +11 2
26. 147 − 24 3
€
€
€
€
area = 36 square inches
€ 12
29. x 11
31.
33. r
36.
6
3
30. a
€
€
€
x 10
y
32.
12
1
34. x 1 10
€
35.
15
6
€
€
3−3 5
38.
2
37.
€
1
12
€
28. perimeter = 18 3 inches
4 7
a 2c 3
12. not a real number
€
20. 2x
€
€
€
€
24. 19x 3 xy 2
23. 20 5x
27. 114
11. −6
1000
2197
5
€
10. 8
1
14.
5
€
€
1
x
Answer Key –€Review Sheet – Math 096 – Final Exam
4 6 9
€
d. f ( x ) =
b
13
1
t
32
€3
or b −1
39. −4 14 − 4 21
€
40.
€
€
€
€
y2
41. x ( x + 25)
44.
€
3 7y
47.
(m −11)(m +11)
50. 3( k +1)( k + 6)
53. 2a ( a +16b)( a + 2b)
€
56. 2 y 3 (3y − 2)( y − 3)
59. x ≠ 7
63.
€
49.
( z + 8)( z − 2)
€
(k −10)(k + 2)
€
€
57.
(
a
7b
6
3z + 4
5z + 3
z
3
70.
€
71.
€
5b 2 + 4b + 8
75.
b( b + 2)
€
€
79.
1
6
€
)
€
81.
{−4,7}
€
58. x (7x −1)(7x +1)
€
€−
61. y ≠ 0, y ≠
m+5
m −5
€65.
−1
1+ 6a
68.
2x + 3
4
69.
x −7
x +7
x +8
4x
72.
€
t +7
€
4 −t
2x
x −3
77.
17
12x
−3
x +4
€
€
85.
{−3 +
62. k ≠ −11, k ≠ 2
€
14, −3 − 14
74.
10 y − 20
( y + 5)( y − 5)
78.
−1
( x + 5)( x +1)
€
⎧
3 ⎫
82. ⎨ −4, ⎬
⎩ 2 ⎭
⎧⎪1 − 2 7 1+ 2 7 ⎫⎪
⎬
,
84. ⎨
⎪⎩ 5
5 ⎪⎭
1
2
€ 2
66.
3
73.
x 2 + 5x −10
76.
( x − 5)( x + 5)( x + 3) €
€
80.
55. 4(2m + 5)( m − 4)
€
(a − 2)(a + 2) a 2 + 4
64.
€
52. (2x −1)(4x + 7)
€
54. (5x − 2)(2x − 3)
60.€Always defined
€
€
( y − 3)( y + 3)( y + 6)
€
67.
€
46.
51. (3x +1)(2x + 5)
€
€
€
45. (4 y −15t )(4 y +15t )
48.
€
)
43. 5z 3 3z5 + 5z 3 +1
€
( x − 5)( x − 2)
(
42. 4x 3 y 2 ( x − 3y )
}
€
€
€
€
83.
{−
86.
{4 + 3
7, 7
}
2, 4 − 3 2
}
⎧⎪ −1+ 65 −1 − 65 ⎫⎪
⎬
,
87. ⎨
⎪⎩
⎪⎭
4
4
⎧⎪ 4 + 6 4 − 6 ⎫⎪
⎬
,
88. ⎨
⎪⎩ 5
5 ⎪⎭
⎧⎪ −2 + 2 7 −2 − 2 7 ⎫⎪
⎬
,
90. ⎨
⎪⎩
⎪⎭€
3
3
€
93.
{−4,10}
96.
{−2,2}
€
€
99.
91.
€
€
{4}
€
{5 +
14,5 − 14
89.
}
102.
{7}
105.
{8}
⎪⎧ −2 + 11 −2 − 11 ⎪⎫
⎬
,
94. ⎨
⎪⎩
⎪⎭ €
2
2
⎪⎧1+ 13 1 − 13 ⎪⎫
⎬
,
95. ⎨
⎪⎩ 2
2 ⎪⎭
97. {10}
98. {−3,1}
€
101.
100. {19}
€
€
103. {16}
(note: 3 is extraneous)
112. 9 inches
104.
{−2,4}
107. {11}
€
π 2 N 2r
109. a = 4€
€
€
106. {7}
€
€ 108. F = Mn
r2
{−4} (note: 1 is extraneous)
€
€
€
⎧⎪ 9 + 241 9 − 241 ⎫⎪
⎬
,
92. ⎨
⎪⎩ 10
10 ⎪⎭
€
€
€
{0,3}
113. 5 feet from house
€
rFkw
110. v =€±
kw
111. r = ±
114. height = 5 inches,
base = 18 inches
€
115. width = 9 meters, length = 17 meters
116. Shorter leg = 6 cm
117. a. 191
b. 11
c. −5
d. z 2 − 8z +11
118. a. 7
b. 19
c. 5
d. −4 p − 4t + 7
5
119. a.
2
1
b.
7
€
Vπh
πh
€
−2x +1
2x −1
c.
or
−3x€− 4
3x + 4
d.
2a 2 +1
3a 2 − 4
€
€
€
€
120. domain = {−7, −4,0,1,3,5} , range = {−2, −1,4,6,8} , it is a function since every x-value
goes to exactly one y-value
121. domain = {2,3,4} , range = {5,7,9,11} , not a function since 2 corresponds to two values in
€
€
the range
€
€
122. domain = {0,1,2,3,4} , range = {0,1,4,9,16} , it is a function since every x-value goes to
exactly one y-value
123. domain = { A, B,C, D, E} , range = {t, u,v, w} , it is a function since every domain value
€
€
goes to exactly one range value
124. domain = [ −5, ∞ ) , range = ( −∞, ∞ ) , not a function since the graph fails the vertical line test
€
€
125. domain = ( −∞, ∞ ) , range = ( −∞, ∞ ) , it is a function since the graph passes the vertical line
test €
€
126. domain = [ −3,3] , range = [ −6,6] , not a function since it fails the vertical line test
€
€
127. domain = ( −∞, ∞ ) , range = [0, ∞) , it is a function since it passes the vertical line test
€
128. a. f (2) = 5
€
€
129
d.
€
b. x = 1 or x = 3
€
€
A. a.
b.
e.
c.
f.