Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If n is odd, under what condition is na negative?
1)
A)
When a is positive
B)
When a is zero
C)
Never
D)
When a is negative
2)
If a > 0, which of the following correctly describes – – a ?
2)
A)
Negative
B)
Not real
C)
Zero
D)
Positive
3)
If a < 0, which of the following correctly describes – – a ?
3)
A)
Positive
B)
Zero
C)
Negative
D)
Not real
4)
If n is even, under what condition is na negative?
4)
A)
When a is zero
B)
Never
C)
When a is negative
D)
When a is positive
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide the proper response.
5)
To rationalize the denominator of an expression such as 5
2, we multiply both the
numerator and the denominator by 2. By what number are we actually multiplying the
given expression, and what property of real numbers justifies the fact that our result is
equal to the given expression?
5)
6)
True or false? A2= A if A is negative.
6)
7)
Can x + 2 = – x have a solution?
7)
8)
Explain how radical denominators are rationalized.
8)
9)
Is 11 simplified?
9)
Provide an appropriate response.
10)
Tell whether the statement is true or false. If it is false, give the correct answer. Explain
your answer. The easiest way to simplify the expression 3
10 –5 is to multiply both the
numerator and denominator by 10 –5.
10)
Provide the proper response.
11)
Is (–2)(–8) = 4?
11)
Provide an appropriate response.
12)
Describe some situation in which it would be important to be able to simplify radical
expressions.
12)
Provide the proper response.
13)
Is 36 =6 ?
13)
Provide an appropriate response.
14)
What are conjugate expressions? Give an example.
14)
15)
Tell whether the statement is true or false. If it is false, give the correct answer. Explain
your answer. The easiest way to simplify the expression 8
2–2 is to multiply both the
numerator and denominator by 2+2.
15)
Provide the proper response.
16)
Explain why A2 A when A is negative.
16)
17)
True or false? 4+25 =4+25
17)
Provide an appropriate response.
18)
Summarize the guidelines to follow when simplifying radical expressions.
18)
Provide the proper response.
19)
A student tries to solve x +7=x+10 by squaring both sides and gets x +7= x +100.
Why is this not correct?
19)
20)
True or false? A2= – A if A is negative.
20)
21)
If p is a prime number, is p in simplified form? Explain.
21)
Provide an appropriate response.
22)
What are the common errors one might make in simplifying radical expressions?
22)
Provide the proper response.
23)
True or false? 3A3= A if A is negative.
23)
24)
Give the definition or an example of the following word or phrase: Like radicals
24)
25)
Explain why the equation x2=25 has two real number solutions, while the equation
x=5 has only one real number solution.
25)
Provide an appropriate response.
26)
Is there any way you can think of to check whether or not your simplified expression is
equivalent to the original radical expression?
26)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the product and quotient rules, as necessary, to simplify the radical expression.
27)
7
3·21
27)
A)
7
B)
49
C)
7
D)
21
Find all square roots of the number.
28)
144
529
28)
A)
–1
2, 1
2
B)
–12
23 , 12
23
C)
–13
23 , 13
23
D)
–3
11 , 3
11
Simplify.
29)
(10 m+ 7 )2
29)
A)
100m2+ 140 m+ 49
B)
10 m2+ 140 m+ 49
C)
100m+ 140 m+ 49
D)
100m+ 49
Find the perimeter of the figure.
30)
A rectangular coal bin has a length of 45 feet and a width of 45 feet. What is its perimeter?
30)
A)
12 5 feet
B)
12 3 feet
C)
6 5 feet
D)
6 3 feet
Simplify.
31)
(8x + 3 2)2
31)
A)
64x2+18
B)
64x2+ 48x 2 +36
C)
64x2+ 48x 2 +18
D)
64x2+36
Provide an appropriate response.
32)
If a has exactly one real square root, what must be true of a?
32)
A)
a is positive
B)
a is nonnegative
C)
a is negative
D)
a is equal to zero
Simplify.
33)
4
125 ·1
25
33)
A)
2 5
25
B)
2
5
C)
2 5
125
D)
2 5
5
Write the quotient in lowest terms.
34)
3 7 – 9
3
34)
A)
7– 3
B)
–2
C)
7+ 3
D)
3 7 – 3
Rationalize the denominator. Assume that all variables represent positive real numbers.
35)
16p5
r
35)
A)
4 p5r
r
B)
4p2p
r
C)
4p2pr
r
D)
4p2p
r
Simplify and add or subtract wherever possible.
36)
2– 2 72 + 5 98
36)
A)
3172
B)
3 2
C)
24 172
D)
24 2
Solve the equation.
37)
q + 4 =4
37)
A)
{16}
B)
{64}
C)
{20}
D)
{12}
Solve the problem.
38)
The product of 32 and the square root of some number is equal to that number decreased by 2048 .
Find the number.
38)
A)
8192
B)
3072
C)
2048
D)
4096
Simplify the expression.
39)
4 5 ( 11 +5 )
39)
A)
455 +5
B)
455 +20
C)
20 11 +20
D)
411 +5
Write the quotient in lowest terms.
40)
48 +54 200
42
40)
A)
24 +54 200
7
B)
24 +270 2
14
C)
8+90 2
7
D)
8+9200
7
Solve the equation.
41)
4y +12 = y
41)
A)
{4 }
B)
{6 , –2}
C)
{6 }
D)
Simplify.
42)
2
9·1
7
42)
A)
126
63
B)
14
21
C)
1
3
2
7
D)
2
21
Solve the equation.
43)
2x + 3 +4 – x = 4
43)
A)
{3}
B)
3, 11
9
C)
D)
{–3}
D)
44)
20x +20 = x + 6
44)
A)
{4, –4}
B)
{4}
C)
{4, 5}
D)
{–4}
D)
Find the product. Assume all variables have non–negative values.
45)
(8 +7 5)(8–7 5)
45)
A)
29
B)
–181
C)
64–49 5
D)
309 +112 5
D)
Simplify the expression.
46)
(3 –2 5)(2– 2 5)
46)
A)
106 – 10 5
B)
6– 6 5
C)
16
D)
26 – 10 5
D)
Solve the equation.
47)
7q – 6 =6
47)
A)
B)
{6}
C)
30
7
D)
{36}
D)
Simplify the radical. Assume that all variables represent nonnegative real numbers.
48)
64m21n6
48)
A)
8m11n4
B)
8m10n3
C)
8m10n3m
D)
8m9n3m
Perform the indicated operation. Assume that all variables represent nonnegative real numbers.
49)
54m5+ 6 481m5
49)
A)
23 4m5
B)
23m4m
C)
491m4m
D)
23 m44m
Rationalize the denominator. Assume that all variables represent positive real numbers.
50)
–27x3
y5
50)
A)
–3x 3xy
y3
B)
–x27xy
y3
C)
–3x
y2
3x
y
D)
–3x33
y5
Find the product. Assume all variables have non–negative values.
51)
(5 6 –4 5)(5 6 +4 5)
51)
A)
25 6–16 5
B)
10
C)
230 –40 30
D)
70
Find the perimeter of the figure.
52)
Find the perimeter of a triangle with sides measuring 432, 250, and 520 meters.
52)
A)
36 2 meters
B)
26 2+ 10 meters
C)
26 2+ 10 5 meters
D)
26 + 10 5 meters
Find the product and simplify.
53)
50 ·72
53)
A)
30 2
B)
60
C)
72
D)
30
Find the product. Assume all variables have non–negative values.
54)
( m +n)( m–n)
54)
A)
m – n
B)
m + n
C)
m – 2 mn + n
D)
m2–n2
A
Simplify the radical. Assume that all variables represent nonnegative real numbers.
55)
c8d12
55)
A)
c4d6
B)
c8d12
C)
c4d6
D)
c4d12
A
Find the root.
56)
5–243
56)
A)
–48.6
B)
–3
C)
Not a real number
D)
3
B
Rationalize the denominator.
57)
5
3+4
57)
A)
15 –4 5
–13
B)
315 +3 5
12
C)
15 +4 5
–13
D)
15 –4 5
7
A
B
Find the perimeter of the figure.
58)
910
210 510
13 10
58)
A)
16 20
B)
24 10
C)
29 20
D)
29 10
Find the length of the unknown side of the right triangle with legs a and b and hypotenuse c. Round to three decimal
places, if necessary.
59)
a =15 , c =39
59)
A)
b =27
B)
b =37
C)
b =36
D)
b =42
Find the indicated root. Give a decimal approximation if necessary. Round to the nearest thousandth.
60)
0.000197
60)
A)
0.014
B)
0.001
C)
1.400
D)
0.140
Simplify.
61)
(8 + 3 3)2
61)
A)
145 + 48 3
B)
145
C)
91 + 48 3
D)
91
11
Find the length of the third side of the right triangle. Give an exact answer and an approximation to three decimal places.
62)
6 c
19
62)
A)
c =397; c =198.5
B)
c =25; c 5.000
C)
c =397
D)
c =397; c 19.925
Simplify.
63)
11
3·11
4
63)
A)
11 3
6
B)
11
6 3
C)
3
611
D)
11 3
2
Solve the problem. Give the answer to the nearest thousandth if necessary.
64)
The diagram below shows a rope connecting the top of a pole to the ground. The rope is 17 yd long
and touches the ground 13 yd from the pole. How tall is the pole?
?17 yd
13 yd
64)
A)
60 yd
B)
5.477 yd
C)
15 yd
D)
10.954 yd
Simplify the radical.
65)
4162
65)
A)
342
B)
9
C)
3
D)
346
Find the square root.
66)
–225
49
66)
A)
15
7
B)
–15
7
C)
–14
3
D)
Not a real number
67)
–0.0049
67)
A)
Not a real number
B)
–0.07
C)
0.07
D)
–0.007
Simplify.
68)
( 3a +b)2
68)
A)
9a2+3ab +b2
B)
3a +3ab +b
C)
9a2+ 2 3ab +b2
D)
3a + 2 3ab +b
Simplify the expression.
69)
3·27
69)
A)
3
B)
1
3
C)
9
D)
3 3
Perform the indicated operation. Assume that all variables represent nonnegative real numbers.
70)
59x10y– 4x 9xy
70)
A)
2x 9xy
B)
59xy – 2x 9xy
C)
Can’t simplify
D)
x9xy
Simplify and add or subtract wherever possible.
71)
10 +410
71)
A)
410
B)
520
C)
510
D)
40
C
Identify the given number as “Rational”, “Irrational”, or “Not a real number”. If the number is rational, give its exact
value. If the number is irrational, give a decimal approximation to the nearest thousandth.
72)
148
72)
A)
Not a real number
B)
Irrational, 12.166
C)
Irrational, 74.000
D)
Rational, 12.166
B
Find the length of the unknown side of the right triangle with legs a and b and hypotenuse c. Round to three decimal
places, if necessary.
73)
a =12 , b =16
73)
A)
c =16
B)
c =14
C)
c =19
D)
c =20
D
Solve the equation.
74)
2x + 15 – x = 6
74)
A)
{–7, –3}
B)
C)
{–3}
D)
{–7}
C
14
D
75)
41 – 2x + 3 =9 – 2x + 7
75)
A)
55
2
B)
7
2
C)
5
2
D)
76)
8a – 5 –5a + 9 = 0
76)
A)
14
3
B)
14
13
C)
3
14
D)
4
3
Simplify the expression.
77)
(11 +154) ( 14 +7)
77)
A)
154 +77 +14 11 +711
B)
154 +77 +14 11 +22
C)
154 +77 +14 11 +722
D)
154 +77 +711 +722
Simplify.
78)
( 2 –10)2
78)
A)
14 – 2 5
B)
12 – 2 5
C)
12 – 4 5
D)
14 +12
79)
17
20 ·20
14
79)
A)
14 17
B)
17
14
C)
17
14
D)
238
14
Find the indicated root. Give a decimal approximation if necessary. Round to the nearest thousandth.
80)
1.81
80)
A)
1.332
B)
1.360
C)
1
D)
1.345
Solve the equation.
81)
10x –18 =3 x
81)
A)
{6}
B)
{18}
C)
{3}
D)
{1}
Rationalize the denominator.
82)
3
7 +6
82)
A)
21 –3 6
6
B)
21 –3 6
43
C)
21 +3 6
43
D)
3
7 – 3
6
Provide an appropriate response.
83)
How many real square roots does any positive number have?
83)
A)
Two
B)
One
C)
It depends on the number
D)
None
Simplify.
84)
31080m7
84)
A)
30m2
B)
6m235m
C)
6m23m
D)
6m235
85)
( w – 10 )2
85)
A)
w+ 100
B)
w– 10 w+ 100
C)
w– 20 w+ 100
D)
w2– 20 w+ 100
Rationalize the denominator.
86)
6–7
6+7
86)
A)
29 –12 7
43
B)
43 –12 7
29
C)
1
D)
43 +12 7
29
Solve the equation.
87)
2k + 1 =9
87)
A)
{80}
B)
{–40}
C)
{20}
D)
{40}
Perform the indicated operation. Assume that all variables represent nonnegative real numbers.
88)
5x +645x+520x
88)
A)
29 5x
B)
11 70x
C)
28 5x
D)
12 70x
17
Provide the proper response.
89)
Which one of the following would be an appropriate choice for multiplying the numerator and the
denominator of
32
37
in order to rationalize the denominator?
89)
A)
32
B)
49
C)
349
D)
37
Perform the indicated operation. Assume that all variables represent nonnegative real numbers.
90)
74x7–3x 4x3
90)
A)
4x 4x7
B)
10 4x3
C)
4x 4x3
D)
74x7–3x 4x3
Solve the problem.
91)
A triangle has a base of 164 and a height of 95. What is the best estimate for its area?
91)
A)
702
B)
58.5
C)
60
D)
65
Simplify the radical.
92)
56
92)
A)
7
B)
14 2
C)
214
D)
28