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Checkpoint 2: Assess Your Understanding, pages 136–138
2.3
1. Group the radicals into sets of like radicals. Simplify first if
necessary.
√ √ √
√ √
√
50, 27, 48, 3 72, 98, 2 3
√
√
√
√
50 ⴝ 25 # 2
27 ⴝ 9 # 3
√
√
ⴝ5 2
ⴝ3 3
√
√
√
√
3 72 ⴝ 3 36 # 2
98 ⴝ 49 # 2
√
√
ⴝ 18 2
√
48 ⴝ
√
16 # 3
√
ⴝ4 3
√
2 3
ⴝ7 2
All radicals have index 2.
√
√
√ √
√ √
√
√ √
50 , 3 72, 98
√ √
√
Radicals with radicand 2: 5 2, 18 2, 7 2; that is,
Radicals with radicand 3: 3 3, 4 3, 2 3; that is,
27 ,
48 , 2 3
2. Multiple Choice Which list contains only like radicals?
Assume all variables are non-negative.
√
√
√
√ √ √
A. 25x, 25x2, 3 25x
B. 3 x, 3 y, 3 a
√
√ √
√
C. 3 x4, 5 x4, 3 5x4
√
√
D. 3 x5, 2x 3 x2, - 3 3 8x5
3. Identify the values of the variables for which each radical is defined,
then simplify.
√
√
√
a) 25x + 36x - 4x
The radicands cannot be negative, so x
» 0.
√
√
√
25x ⴙ 36x ⴚ 4x ⴝ 5 x ⴙ 6 x ⴚ 2 x
√
√
√
√
ⴝ9 x
b)
√
√
√
√
8a - 3 b + 5 2a + 4b
» 0 and b » 0.
√
√
√
√
8a ⴚ 3 b ⴙ 5 2a ⴙ 4b ⴝ 2 2a ⴚ 3 b ⴙ 5 2a ⴙ 2 b
√
√
The radicands cannot be negative, so a
√
√
√
√
ⴝ 7 2a ⴚ
©P
DO NOT COPY.
b
Chapter 2: Absolute Value and Radicals—Checkpoint 2—Solutions
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c)
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√
√
√
√
75c4d + c2 12d + 48cd4 - 3d2 27c
√
√
12d » 0, so d » 0 and 27c » 0, so c » 0
√ 4
√
√
√
75c d ⴙ c2 12d ⴙ 48cd4 ⴚ 3d2 27c
√
√
√
√
ⴝ 25 # 3 # c4 # d ⴙ c2 4 # 3d ⴙ 16 # 3 # c # d4 ⴚ 3d2 9 # 3 # c
√
√
√
√
ⴝ 5c2 3d ⴙ 2c2 3d ⴙ 4d2 3c ⴚ 9d2 3c
√
√
2
2
ⴝ 7c
3d ⴚ 5d
3c
√
√
√
d) 3 8a + 3 16a4 - 3 -128a
The cube root of a number is defined for all real numbers. So, each
radical is defined for a ç ⺢.
√
3
8a ⴙ
√3
16a4 ⴚ
√
3
√
√
√
3
8 # a ⴙ 3 8 # 2 # a3 # a ⴚ 3 ⴚ64 # 2 # a
√
√
√
ⴝ 2 3 a ⴙ 2a 3 2a ⴚ (ⴚ4) 3 2a
√
√
√
3
3
3
ⴚ128a ⴝ
ⴝ 2 a ⴙ 2a 2a ⴙ 4 2a
2.4
4. Expand and simplify.
√
√
a) 1 7 + 222
√
√ √
√
ⴝ ( 7 ⴙ 2)( 7 ⴙ 2)
√ √
√
ⴝ 7( 7 ⴙ 2)
√ √
√
ⴙ 2( 7 ⴙ 2)
√
√
ⴝ 7 ⴙ 14 ⴙ 14 ⴙ 2
√
√
√
b) 1 7 - 222
√
√ √
√
ⴝ ( 7 ⴚ 2)( 7 ⴚ 2)
√ √
√
√ √
√
ⴝ 7( 7 ⴚ 2) ⴚ 2( 7 ⴚ 2)
√
√
ⴝ 7 ⴚ 14 ⴚ 14 ⴙ 2
√
√ √
√
√
c) 1 7 + 221 7 - 22
√ √
√
ⴝ 7( 7 ⴚ 2)
√ √
√
ⴙ 2( 7 ⴚ 2)
√
√
√
√ √
√
d) 12 7 + 3 221 7 - 2 22
√ √
√
√ √
√
ⴝ 2 7( 7 ⴚ 2 2) ⴙ 3 2( 7 ⴚ 2 2)
√
√
ⴝ 14 ⴚ 4 14 ⴙ 3 14 ⴚ 12
√
ⴝ 9 ⴚ 2 14
ⴝ 9 ⴙ 2 14
ⴝ7ⴚ
14 ⴙ
14 ⴚ 2
ⴝ2ⴚ
14
ⴝ5
5. Multiple Choice Which expression represents
√
√
√
√
√
√
13 x - 2 y213 x + 2 y2 - 12 x - 3 y22, x ≥ 0, y ≥ 0,
in simplest form?
A. 5x + 5y + 12xy
C. 5x - 13y + 12xy
30
√
B. 5x - 5y + 12 xy
√
D. 5x - 13y + 12 xy
Chapter 2: Absolute Value and Radicals—Checkpoint 2—Solutions
DO NOT COPY.
©P
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6. Rationalize the denominator.
√
√
√
8 3 + 1
3 24 - 4 2
√
a) √
b)
3
2
√
√ √
√
√
(6 6 ⴚ 4 2) # 2
(8 3 ⴙ 1) # 3
√
√
ⴝ
√
√
ⴝ
3
√
ⴝ
3
# √3 ⴙ 1 # √3
√ √
3# 3
8 3
24 ⴙ
ⴝ
3
√
√
ⴝ
6 6
√
2
2
# √2 ⴚ 4√2 # √2
√ √
2# 2
6 12 ⴚ 8
ⴝ
2
3
√
12 3 ⴚ 8
√2
2(6 3 ⴚ 4)
ⴝ
2
ⴝ
√
ⴝ6 3ⴚ4
7. Simplify.
√
3 2
a) √
2 6 - 5
√
3 2
√
# (2√6 ⴙ 5)
√
(2 6 ⴚ 5) (2 6 ⴙ 5)
√
√
6 12 ⴙ 15 2
ⴝ √ 2
(2 6) ⴚ (5)2
√
√
ⴝ
ⴝ
12 3 ⴙ 15 2
24 ⴚ 25
√
√
12 3 ⴙ 15 2
ⴝ
ⴚ1
√
√
ⴝ ⴚ12 3 ⴚ 15 2
√
√
3 8 + 2 5
√
b) √
2 + 20
√
√
6 2ⴙ2 5
ⴝ √
√
2ⴙ2 5
√
√
√
√
(6 2 ⴙ 2 5) # ( 2 ⴚ 2 5)
√
√
√
ⴝ √
( 2 ⴙ 2 5) ( 2 ⴚ 2 5)
√ √
√
√ √
√
6 2( 2 ⴚ 2 5) ⴙ 2 5( 2 ⴚ 2 5)
√
√
ⴝ
( 2)2 ⴚ (2 5)2
√
√
ⴝ
ⴝ
12 ⴚ 12 10 ⴙ 2 10 ⴚ 20
2 ⴚ 20
√
ⴚ 8 ⴚ 10 10
ⴚ18
√
4 ⴙ 5 10
ⴝ
9
©P
DO NOT COPY.
Chapter 2: Absolute Value and Radicals—Checkpoint 2—Solutions
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