Download Simplify each expression. 2. SOLUTION: 4. SOLUTION: 6

SOLUTION: 10-2 Simplifying Radical Expressions
Simplify each expression.
10. MULTIPLE CHOICE Which expression is
2. equivalent to
?
SOLUTION: A B C 4. D SOLUTION: SOLUTION: 6. SOLUTION: The correct choice is D.
8. Simplify each expression.
SOLUTION: 12. SOLUTION: 10. MULTIPLE CHOICE Which expression is
equivalent to
?
A B 14. C SOLUTION: D SOLUTION: eSolutions Manual - Powered by Cognero
Page 1
10-2 Simplifying Radical Expressions
22. 14. SOLUTION: SOLUTION: 24. SOLUTION: 16. 26. SOLUTION: SOLUTION: 28. SOLUTION: Simplify each expression.
18. SOLUTION: 30. SOLUTION: 20. SOLUTION: 32. SOLUTION: 22. SOLUTION: eSolutions
Manual - Powered by Cognero
Page 2
34. SOLUTION: SOLUTION: Therefore, this pump will not meet the fire
department’s need.
c. To determine the height of the water, substitute 77
10-2 Simplifying Radical Expressions
for v in the function
.
34. SOLUTION: A pump with a velocity of 77 feet per second will
pump water to a maximum height of 92.6 feet.
Therefore, this pump will meet the fire department’s
need.
36. CCSS PRECISION When fighting a fire, the
velocity v of water being pumped into the air is
modeled by the function
, where h
represents the maximum height of the water and g
2
represents the acceleration due to gravity (32 ft/s ).
a. Solve the function for h.
b. The Hollowville Fire Department needs a pump
that will propel water 80 feet into the air. Will a pump
advertised to project water with a velocity of 70 feet
per second meet their needs? Explain.
c. The Jackson Fire Department must purchase a
pump that will propel water 90 feet into the air. Will a
pump that is advertised to project water with a
velocity of 77 feet per second meet the fire
department’s need? Explain.
Simplify each expression.
38. SOLUTION: SOLUTION: a.
40. SOLUTION: b. To determine the height of the water, substitute 70
for v in the function
.
42. A pump with a velocity of 70 feet per second will
pump water only to a maximum height of 76.6 feet.
Therefore, this pump will not meet the fire
department’s need.
c. To determine the height of the water, substitute 77
for v in the function
SOLUTION: .
eSolutions Manual - Powered by Cognero
Page 3
10-2 Simplifying Radical Expressions
42. 48. SOLUTION: SOLUTION: 50. KINETIC ENERGY The speed v of a ball can be
determined by the equation
44. SOLUTION: , where k is the
kinetic energy and m is the mass of the ball.
a. Simplify the formula if the mass of the ball is 3
kilograms.
b. If the ball is traveling 7 meters per second, what
is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m
= 3 into the equation.
46. SOLUTION: b. Substitute V = 7 in the equation
.
48. SOLUTION: eSolutions Manual - Powered by Cognero
50. KINETIC ENERGY The speed v of a ball can be
Page 4
The kinetic energy of the ball is 73.5 Joules.
10-2 Simplifying Radical Expressions
The kinetic energy of the ball is 73.5 Joules.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation
52. CCSS STRUCTURE Explain how to solve
.
, where k is the
kinetic energy and m is the mass of the ball.
a. Simplify the formula if the mass of the ball is 3
kilograms.
b. If the ball is traveling 7 meters per second, what
is the kinetic energy of the ball in Joules?
SOLUTION: To solve an equation of equal ratios, first find the
equal cross products and then solve for the variable.
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m
= 3 into the equation.
Use the conjugate of denominator.
b. Substitute V = 7 in the equation
to rationalize the .
So, the solution is
.
54. REASONING Marge takes a number, subtracts 4,
multiplies by 4, takes the square root, and takes the
reciprocal to get
. What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
The kinetic energy of the ball is 73.5 Joules.
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the
equal cross products and then solve for the variable.
eSolutions Manual - Powered by Cognero
56. CHALLENGE Use the Quotient Property of Page 5
Square Roots to derive the Quadratic Formula by
2
solving the quadratic equation ax + bx + c = 0.
g – 23.
The two bills are a total of $109.
10-2 Simplifying Radical Expressions
So, the correct choice is D.
56. CHALLENGE Use the Quotient Property of
Square Roots to derive the Quadratic Formula by
2
solving the quadratic equation ax + bx + c = 0.
(Hint: Begin by completing the square.)
60. The expression
the following?
A is equivalent to which of B SOLUTION: C D SOLUTION: So, the correct choice is C.
58. Jerry’s electric bill is $23 less than his natural gas bill.
The two bills are a total of $109. Which of the
following equations can be used to find the amount of
his natural gas bill?
A g + g = 109
B 23 + 2g = 109
C g − 23 = 109
D 2g − 23 = 109
Graph each function. Compare to the parent
graph. State the domain and range.
62. SOLUTION: x
0
1
y
1
–1
2
≈ 1.8
3
≈ 2.5
4
3
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is
$23 less than his natural gas bill, so the electric bill is
g – 23.
The two bills are a total of $109.
So, the correct choice is D.
60. The expression
the following?
A is multiplied by a value
The parent function greater than 1 and is subtracted by the value 1, so the
graph is a vertical stretch of
followed by a
translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ -1}.
is equivalent to which of B C D SOLUTION: eSolutions Manual - Powered by Cognero
64. SOLUTION: x
–2
–1
y
0
2
0
≈ 2.8
1
≈ 3.5
2
4
Page 6
greater than 1 and is subtracted by the value 1, so the
graph is a vertical stretch of
followed by a
translation 1 unit down. The domain is {x|x ≥ 0}, and 10-2the
Simplifying
Radical Expressions
range is {y|y ≥ -1}.
Determine the domain and range for each
function.
68. f (x) = |2x − 5|
64. SOLUTION: x
–2
–1
y
0
2
This graph is the result of a vertical stretch of the
graph of
followed by a reflection across the
x-axis, and then a translation 3 units right. The
domain is {x|x ≥ 3}, and the range is {y|y ≤ 0}.
0
≈ 2.8
1
≈ 3.5
2
4
SOLUTION: Since f (x) cannot be negative, the minimum point of
the graph is where f (x) = 0.
Make a table of values.
x
0
1
f (x)
5
3
This graph is the result of a vertical stretch of the
graph of y =
followed by a translation 2 units
left. The domain is {x|x ≥ –2}, and the range is {y|y
≥ 0}.
2
1
3
1
4
3
66. SOLUTION: x
3
4
y
0
–3
5
≈ –
4.2
6
≈ –
5.2
The domain is all real numbers, and the range is {y |
y ≥ 0}.
70. This graph is the result of a vertical stretch of the
graph of
followed by a reflection across the
x-axis, and then a translation 3 units right. The
domain is {x|x ≥ 3}, and the range is {y|y ≤ 0}.
Determine the domain and range for each
function.
68. f (x) = |2x − 5|
SOLUTION: Since f (x) cannot be negative, the minimum point of
the graph is where f (x) = 0.
eSolutions Manual - Powered by Cognero
SOLUTION: This is a piecewise-defined function. Make a table
of values. Be sure to include the domain values for
which the function changes.
x
-1
0
1
2
3
g(x)
-2
-1
0
1
-5
Notice that both functions are linear.
Page 7
The domain is all real numbers, and the range is {y |
y ≥ 0}.
10-2 Simplifying Radical Expressions
There are no real positive square roots of –100.
Therefore, the solution to this equation is ø.
2
74. 2r + r − 14 = 0
70. SOLUTION: SOLUTION: For this equation, a = 2, b = 1, and c = –14.
This is a piecewise-defined function. Make a table
of values. Be sure to include the domain values for
which the function changes.
x
-1
0
1
2
3
g(x)
-2
-1
0
1
-5
Notice that both functions are linear.
The solutions are –2.9 and 2.4.
2
76. 11z − z = 3
The domain is all real numbers, and the range is {y |
y ≤ 1}.
Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 11, b = –1, and c = –3.
2
72. r + 25 = 0
SOLUTION: For this equation, a = 1, b = 0, and c = 25.
The solutions are −0.5 and 0.6.
There are no real positive square roots of –100.
Therefore, the solution to this equation is ø.
Factor each polynomial, if possible. If the
polynomial cannot be factored, write prime .
78. 4 − 9a
2
74. 2r + r − 14 = 0
2
SOLUTION: SOLUTION: For this equation, a = 2, b = 1, and c = –14.
eSolutions Manual - Powered by Cognero
Page 8
4
80. 32x − 2y
4
10-2 Simplifying Radical Expressions
The solutions are −0.5 and 0.6.
Factor each polynomial, if possible. If the
polynomial cannot be factored, write prime .
78. 4 − 9a
2
tomatoes: t + 200 > 10,000.
Solving, we get: t > 10,000 - 200
t > 9,800
Write the prime factorization of each number.
86. 88
SOLUTION: SOLUTION: 88. 31
4
80. 32x − 2y
4
SOLUTION: SOLUTION: The number 31 is prime. So, the prime factorization
of 31 is 31.
90. 90
SOLUTION: 3
2
82. x − 3x − 9x + 27
SOLUTION: 84. TOMATOES There are more than 10,000 varieties
of tomatoes. One seed company produces seed
packages for 200 varieties of tomatoes. For how
many varieties do they not provide seeds?
SOLUTION: Let t be the number of varieties of tomatoes for
which the seed company does not produce. Since the
seed company produces packages for 200 varieties,
the total number of varieties can be expressed as: t +
200. Since there are more than 10,000 varieties of
tomatoes: t + 200 > 10,000.
Solving, we get: t > 10,000 - 200
t > 9,800
Write the prime factorization of each number.
86. 88
SOLUTION: eSolutions Manual - Powered by Cognero
88. 31
Page 9