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Lesson 2.2 Exercises, pages 100–105
A
3. Write each mixed radical as an entire radical.
√
√
√
a) 6 5 ⴝ 36 # 5
√
ⴝ
√
√ √
b) 3 3 4 ⴝ 3 33 # 3 4
√
√
ⴝ 3 27 # 3 4
√3
180
ⴝ
√
√
√
c) -2 3 5 ⴝ 3 (ⴚ 2)3 # 3 5
√
√
3
3
ⴝ
ⴝ
√
3
ⴚ8
#
5
108
√ √
√
d) 5 4 2 ⴝ 4 54 # 4 2
√4
√
4
ⴝ
ⴚ40
ⴝ
625
√4
#
2
1250
4. Write each entire radical as a mixed radical, if possible.
a)
√
√
54 ⴝ 9 # 6
√
b)
ⴝ3 6
c)
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√
3
-81 ⴝ
√3
#
ⴚ27
√ 3
ⴝ ⴚ3 3 3
DO NOT COPY.
√
3
√
96 ⴝ 3 8 # 12
√
3
ⴝ 2 12
d)
√
4
47
√4
47cannot be written as a
mixed radical because 47
does not have any factors that
are perfect fourth powers
2.2 Simplifying Radical Expressions—Solutions
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5. Arrange in order from least to greatest.
√
√ √
a) 6 3 7, 11 3 7, 5 3 7
b)
Each mixed √
radical is a
multiple of 3 7 , so compare
the coefficients: 5 < 6 < 11
So,
√ from
√ least√to greatest:
5 3 7, 6 3 7, 11 3 7
√
√ √
28, 2 5, 3 3
Each radical has index 2.
Write each
√
√ mixed
√ radical
√ as an
√ entire
√ radical.
√
28 2 5 ⴝ 22 # 5 3 3 ⴝ 32 # 3
√
4#5
√
ⴝ
ⴝ
ⴝ
ⴝ
20
√
9#3
√
27
Compare the radicands: 20 <√27 <√28 √
So, from least to greatest: 2 5, 3 3, 28
B
6. For which values of the variable is each radical defined?
Justify your answers.
a)
√
√
18x2
b)
18x2 H ⺢ when 18x2 » 0.
18>
x2 » 0
√ 0 and
So, 18x2 is defined for x H ⺢.
c)
√
-2a3
√
ⴚ2a3 H ⺢ when ⴚ 2a3 » 0.
3
ⴚ2<
√ 0, so3 a ◊ 0; that is, a ◊ 0
So, ⴚ2a is defined for a ◊ 0.
√
3
4c3
d)
Since the cube root of a number
is defined for all real numbers,
√
3
4c3 H ⺢ when 4c3 H ⺢.
√
√
4
√
4
18z5
18z5 H ⺢ when 18z5 » 0.
18 >√0, so z5 » 0; that is, z » 0
So, 4 18z5 is defined for z » 0.
So, 3 4c3 is defined for c H ⺢.
√
√
7. Write each mixed radical as an entire radical.
7
2 5
2
a) 3 3
b) - 3 8
c) - 3
ⴝ
√
32
√
√7
3
9a b
ⴝ
ⴝ
#
√
7
3
21
√
√ 2 √5
#
3
√4 5 8
ⴝ ⴚ a ba b
√ 59 8
2
ⴝ ⴚ a b
ⴝ ⴚ
18
√2
3
3
√ 2 √2
aⴚ b #
√ 38 2 3
ⴝ aⴚ ba b
√ 1627 3
ⴝ
3
3
3
3
ⴝ
√
3
ⴚ
81
√
8. Write each entire radical as a mixed radical, if possible.
125
32
64
a)
b) 4 243
c) 3 - 81
4
√
√
√
4
3
ⴝ
25 # 5
√
4
√
5 5
ⴝ
2
16 # 2
ⴝ√
4
81 # 3
√
ⴝ √
4
24 2
√
3 3
ⴝ√
3
ⴚ 64
27 # 3
ⴚ4
ⴝ √
3
3 3
2 2
ⴝ 4
3 3
8
2.2 Simplifying Radical Expressions—Solutions
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9. Arrange in order from greatest to least.
√ √
√
a) 3 3 5, 2 11, 4 4 8
The√mixed radicals have different indices, so use a calculator.
3 3 5 ⴝ 5.1299 Á
√
√
4
2 11 ⴝ 6.6332 Á
4 8 ⴝ 6.7271 Á
Compare the decimals: 6.7271
√ >√6.6332√>5.1299
So, from greatest to least: 4 4 8, 2 11, 3 3 5
√ √ √ √
b) 4 6, 4 7, 4 2, 4 5
Each mixed radical has index 2 and coefficient 4.
> 5>
So, compare the radicands: 7√>6√
√ 2√
So, from greatest to least: 4 7, 4 6, 4 5, 4 2
√ √
√ √
c) 3 3 5, 2 3 12, 4 3 2, 3 3 4
Each radical has index 3.
Write
mixed radical
entire radical.
√ each
√
√ as an√
3 3 #
3 3 #
335 ⴝ √
3 5
2 3 12 ⴝ √
2 12
3
3
ⴝ
135
ⴝ
96
√
√
√
√
3
3 3 #
3
3 3 #
4 2 ⴝ √4 2
3 4 ⴝ √3 4
ⴝ 3 108
ⴝ 3 128
Compare the radicands: 135√
108>
>128
√ >√
√ 96
So, from greatest to least: 3 3 5, 4 3 2, 3 3 4, 2 3 12
10. Write the values of the variable for which each radical is defined.
Simplify the radical, if possible.
a)
√
3
-8x2
b)
ⴚ8x ⴝ
2
√3
√
ⴚ8 # x
√
3
√
48b4
48b4 ç ⺢ when 48b4 » 0.
b4 » 0
48>
√ 0 and
4
So, 48b is defined for b ç ⺢.
Since the cube root of a
number is defined for all real
values of x, the radical is
defined for x ç ⺢.
√
3
√
2
48b4 ⴝ
√
4
16r4
d)
√
4
16r4 ç ⺢ when 16r4
4
16>0 and r
√
4
»0
» 0.
So, 16r is defined for r ç ⺢.
√
4
4
√
16r4 ⴝ 4 16 # r4
ⴝ 2 r ©P
DO NOT COPY.
16 # 3 # b4
√
ⴝ 4b2 3
ⴝ ⴚ2 x2
c)
√
√
4
125x3
√
4
125x3 ç ⺢ when 125x3
» 0.
3
125>0 so x
» 0; that is, x » 0
So, 125x is defined for x » 0.
√
4
3
√4
3
125x cannot be simplified because
the radicand does not have any factors
that are perfect fourth powers.
2.2 Simplifying Radical Expressions—Solutions
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11. For which values of the variable is each radical defined?
√27x
√ - 48c
Rewrite the radical in simplest form.
a)
2
3
b)
16
Since the cube root of a number
is defined for all real values of x,
the radical is defined for x ç ⺢.
√27x
3
2
16
ⴝ
ⴝ
c)
√27 # x
8#2
√
3 x
3
3
2
2
5
3
125
Since the cube root of a number
is defined for all real values of c,
the radical is defined for c ç ⺢.
√ⴚ 48c
3
125
2
2
5
ⴝ
√ⴚ 8c
3
3
# 6c
2
125
2 √
3
ⴝ ⴚ c 6c2
5
√81y
5
4
16
√81y
ç ⺢ when y
16
√81y
√81y # y
5
4
5
5
4
16
ⴝ
4
» 0; that is, y » 0
4
16
3 √
4
ⴝ y y
2
12. Determine whether each statement is:
• always true
• sometimes true
• never true
Justify your answer.
a) 1 -x22 = x2
b)
Always true; a number multiplied by itself is always positive.
√
x2 = ;x
Sometimes true;
√2
√
x ⴝ x , and x » 0, so when x ⴝ 0, x 2 ⴝ — x .
√ 2
But, when x ⴝ 0,
x ⴝ —x
13. In the far north, ice roads are often the only way to get supplies to
√
mines. The formula t = 4 m is used to determine the minimum
thickness of ice, t inches, needed for an ice road to support a mass of
m tons.
a) What is the minimum thickness of ice needed for a mass of 10 T?
√
Use the
√ formula t ⴝ 4 m. Substitute: m ⴝ 10
t ⴝ 4 10
√
So, the minimum thickness of ice needed is 4 10 in.
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2.2 Simplifying Radical Expressions—Solutions
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b) What is the maximum mass for ice that is 10 in. thick?
√
Use the formula t ⴝ 4 m. Substitute: t ⴝ 10
√
10 ⴝ 4 m
5
ⴝ
2
2
5
a b ⴝ
2
mⴝ
√
m
√
( m )2
25
, or 6.25
4
So, the maximum mass is 6.25 T.
c) Suppose the mass of a load is multiplied by 4. How will the
thickness of the ice have to increase? Explain your thinking.
√
When the mass is multiplied by 4, t ⴝ 4 4m
√
ⴝ4#2 m
√
Compare this to the original equation: t ⴝ 4 m
√
The thickness of the ice will be multiplied by 2, which is
4.
C
14. For which values of the variables is each radical defined?
Simplify the radical, if possible.
a)
√
98a3b6
√
b)
98a3b6 ç ⺢ when 98a3b6
98>0 and b6 » 0
So, a3
√
» 0; that is, a » 0
98a3b6 is defined for
a
√
» 0.
» 0, b ç ⺢.
√
3 6
98a b ⴝ
√
3
-40x3y5
Since the cube root of a number
is defined for all real numbers,
√
3
ⴚ40x3y5 is defined for x ç ⺢, y ç ⺢.
√3
√
ⴚ40x3y5 ⴝ 3 ⴚ8 # 5 # x3y3 # y2
√3 2
ⴝ ⴚ2xy 5y
2 6
49 # 2 # a b # a
√
ⴝ 7ab3 2a
c)
√
4
√
4
√
d) 3 128m2n4
48r4s8
48r4s8 ç ⺢ when 48r4s8 » 0.
48>
0, r4 » 0, and s8 » 0
√
4
So, 48r4s8 is defined
for r ç ⺢, s ç ⺢.
√
4
48r4s8 ⴝ
√
4
Since the cube root of a number
is defined for all real numbers,
√
3
128m2n4 is defined for m ç ⺢, n ç ⺢.
√
3
128m2n4 ⴝ
16 # 3 # r4s8
√
√
3
64 # 2 # n3 # m2n
√
ⴝ 4n 3 2m2n
ⴝ 2 rs2 4 3
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2.2 Simplifying Radical Expressions—Solutions
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15. The surface area of sphere B is A square units. The surface area of
sphere C is twice that of sphere B. The surface area of sphere D is
9 times that of sphere B. Determine the radii of spheres C and D in
terms of A.
B
Formula for surface area of a sphere: SA ⴝ 4␲r2
Sphere B has surface area A square units.
Sphere C
Sphere D
Surface area: 2A
Surface area: 9A
So, 2A ⴝ 4␲r2
So, 9A ⴝ 4␲r2
C
D
2A
ⴝ r2
4␲
9A
ⴝ r2
4␲
√9A
ⴝr
4␲
√
3 A
A
ⴝ r2
2␲
√A
2␲
ⴝr
Sphere C has radius
√A
2␲
ⴝr
2 ␲
units.
Sphere D has radius
√
3 A
units.
2 ␲
16. A circle with radius r and area A
touches another circle with radius
y and area 4A. A larger circle is
drawn around the other two
as shown.
4A
A
r
y
a) Determine an expression for the
value of y in terms of r. State the
restrictions on the variables.
Formula for area of a circle: A ⴝ ␲r2
Since r and y are lengths, r>0 and y>0.
Area of small circle: A ⴝ ␲r2
Area of middle circle: 4A ⴝ ␲y2
To determine y in terms of r, substitute A ⴝ ␲r2 in 4A ⴝ ␲y2, then
solve for y.
4A ⴝ ␲y2
4(␲r2) ⴝ ␲y2
4␲r2
2
␲ ⴝy
4r2 ⴝ y2
√
y ⴝ 4r2
y ⴝ 2r
b) Determine the area of the shaded region in terms of A.
Area of shaded region
ⴝ Area of large circle ⴚ area of 2 smaller circles
ⴝ ␲( y ⴙ 2r)2 ⴚ 4A ⴚ A Substitute: y ⴝ 2r
ⴝ ␲(2r ⴙ 2r)2 ⴚ 5A
ⴝ ␲(4r)2 ⴚ 5A
ⴝ 16␲r2 ⴚ 5A
Substitute: ␲r2 ⴝ A
ⴝ 16A ⴚ 5A
ⴝ 11A
The area of the shaded region is 11A.
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2.2 Simplifying Radical Expressions—Solutions
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