10-2 Simplifying Radical Expressions
Simplify each expression.
1. SOLUTION: 2. SOLUTION: 3. SOLUTION: 4. SOLUTION: 5. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
5. SOLUTION: 6. SOLUTION: 7. SOLUTION: 8. SOLUTION: 9. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
9. SOLUTION: 10. MULTIPLE CHOICE Which expression is equivalent to
?
A B C D SOLUTION: The correct choice is D.
Simplify each expression.
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SOLUTION: Page 3
10-2 Simplifying Radical Expressions
The correct choice is D.
Simplify each expression.
11. SOLUTION: 12. SOLUTION: 13. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
13. SOLUTION: 14. SOLUTION: 15. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
15. SOLUTION: 16. SOLUTION: Simplify each expression.
17. SOLUTION: 18. eSolutions Manual - Powered by Cognero
SOLUTION: Page 6
10-2 Simplifying Radical Expressions
18. SOLUTION: 19. SOLUTION: 20. SOLUTION: 21. SOLUTION: 22. eSolutions Manual - Powered by Cognero
SOLUTION: Page 7
10-2 Simplifying Radical Expressions
22. SOLUTION: 23. SOLUTION: 24. SOLUTION: 25. SOLUTION: 26. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
26. SOLUTION: 27. SOLUTION: 28. SOLUTION: 29. SOLUTION: 30. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
30. SOLUTION: 31. SOLUTION: 32. SOLUTION: 33. SOLUTION: 34. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
34. SOLUTION: 35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at
the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation.
b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the equation
.
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
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 represents the acceleration due to
2
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.
SOLUTION: a.
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10-2 Simplifying Radical Expressions
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
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 represents the acceleration due to
2
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.
SOLUTION: a.
b. To determine the height of the water, substitute 70 for v in the function
.
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
.
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.
Simplify each expression.
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10-2ASimplifying
pump with aRadical
velocityExpressions
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.
Simplify each expression.
37. SOLUTION: 38. SOLUTION: 39. eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
39. SOLUTION: 40. SOLUTION: 41. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
41. SOLUTION: 42. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
42. SOLUTION: 43. SOLUTION: 44. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
44. SOLUTION: 45. SOLUTION: 46. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
46. SOLUTION: 47. SOLUTION: 48. SOLUTION: eSolutions Manual - Powered by Cognero
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10-2 Simplifying Radical Expressions
48. SOLUTION: 49. ELECTRICITY The amount of current in amperes I that an appliance uses can be calculated using the formula
, where P is the power in watts and R is the resistance in ohms.
a. Simplify the formula.
b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed
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v of a ball can be determined by the equation
energy and m is the mass of the ball.
a. Simplify the formula if the mass of the ball is 3 kilograms.
, where k is the kinetic
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10-2 Simplifying Radical Expressions
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be determined by the equation
, 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.
b. Substitute V = 7 in the equation
.
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles that the lookout can see on a clear day is modeled by the formula
. Determine how high the submarine would have to raise its periscope to see a ship, if the submarine is the
given distances away from a ship.
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10-2 Simplifying Radical Expressions
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles that the lookout can see on a clear day is modeled by the formula
. Determine how high the submarine would have to raise its periscope to see a ship, if the submarine is the
given distances away from a ship.
SOLUTION: First solve the equation for h.
Distance
Height =
3
d
2
h=
6
2
(3) = 6 h =
9
2
(6) = 24
h=
12
2
(9) = 54
h=
15
2
(12) = 96
h=
2
(15) = 150
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.
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Distance
3
2
6
2
Height = d h = (3) = 6 h =
10-2 Simplifying Radical Expressions
9
2
(6) = 24
h=
12
2
(9) = 54
h=
15
2
(12) = 96
h=
2
(15) = 150
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.
Use the conjugate of to rationalize the denominator.
So, the solution is
.
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION: a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the reciprocal to
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get
. What number did she start with? Write a formula to describe the process.
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b.
10-2 Simplifying Radical Expressions
c.
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.
55. OPEN ENDED Write two binomials of the form
and . Then find their product.
SOLUTION: Two binomials of the form
and are
and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by solving the
2
quadratic equation ax + bx + c = 0. (Hint: Begin by completing the square.)
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Two binomials of the form
and are
and .
10-2 Simplifying Radical Expressions
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by solving the
2
quadratic equation ax + bx + c = 0. (Hint: Begin by completing the square.)
SOLUTION: 57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in
the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify.
For example, simplify the following.
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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
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10-2 Simplifying Radical Expressions
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in
the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify.
For example, simplify the following.
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
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.
2
59. Solve a − 2a + 1 = 25.
F −4, −6
G 4, −6
H −4, 6
J 4, 6
SOLUTION: Solve for a.
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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.
10-2 Simplifying Radical Expressions
So, the correct choice is D.
2
59. Solve a − 2a + 1 = 25.
F −4, −6
G 4, −6
H −4, 6
J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression
is equivalent to which of the following?
A B C D SOLUTION: So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had
a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
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Graph each function. Compare to the parent graph. State the domain and range.
62. Page 26
10-2 Simplifying Radical Expressions
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had
a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Graph each function. Compare to the parent graph. State the domain and range.
62. SOLUTION: x
0
1
y
1
–1
2
≈ 1.8
The parent function a vertical stretch of
1}.
3
≈ 2.5
4
3
is multiplied by a value greater than 1 and is subtracted by the value 1, so the graph is
followed by a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ -
63. SOLUTION: x
0
1
y
0
2
≈ 0.7
The parent function
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3
≈0 .9
4
1
is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical
compression of
. Another way to identify the compression is to notice that the y-values in the table are
times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
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is multiplied by a value greater than 1 and is subtracted by the value 1, so the graph is
The parent function a vertical stretch of
followed by a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ 10-2 Simplifying Radical Expressions
1}.
63. SOLUTION: x
0
1
y
0
2
≈ 0.7
The parent function
3
≈0 .9
4
1
is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical
compression of
. Another way to identify the compression is to notice that the y-values in the table are
times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
64. SOLUTION: x
–2
–1
y
0
2
0
≈ 2.8
1
≈ 3.5
2
4
This graph is the result of a vertical stretch of the graph of y =
is {x|x ≥ –2}, and the range is {y|y ≥ 0}.
followed by a translation 2 units left. The domain
65. SOLUTION: x
0
–1
y
0
–1
1
≈ –
1.4
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2
≈ –
1.7
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graph is the
result Expressions
of a vertical stretch of the graph of y =
10-2This
Simplifying
Radical
is {x|x ≥ –2}, and the range is {y|y ≥ 0}.
followed by a translation 2 units left. The domain
65. SOLUTION: x
0
–1
y
0
–1
1
≈ –
1.4
2
≈ –
1.7
This graph is the result of a reflection across the x-axis of the graph of
The domain is {x|x ≥ –1}, and the range is {y|y ≤ 0}.
followed by a translation 1 unit left.
66. SOLUTION: x
3
4
y
0
–3
5
≈ –
4.2
6
≈ –
5.2
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}.
67. SOLUTION: x
0
1
y
1
–1
2
≈ –
1.8
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3
≈ –
2.5
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graph is the
result Expressions
of a vertical stretch of the graph of
followed by a reflection across the x-axis, and
10-2This
Simplifying
Radical
then a translation 3 units right. The domain is {x|x ≥ 3}, and the range is {y|y ≤ 0}.
67. SOLUTION: x
0
1
y
1
–1
2
≈ –
1.8
3
≈ –
2.5
is multiplied by a value less than 1 and is added to the value 1, so the graph is a vertical
The parent function stretch of
followed by a reflection across the x-axis, and then a translation 1 unit up. The domain is {x|x ≥ 0}, and the range is {y|y ≤ 1}.
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.
Make a table of values.
x
0
f (x)
5
1
3
2
1
3
1
4
3
The domain is all real numbers, and the range is {y | y ≥ 0}.
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69. h(x) = |x − 1|
SOLUTION: Page 30
is multiplied by a value less than 1 and is added to the value 1, so the graph is a vertical
The parent function of
followed
by a reflection across the x-axis, and then a translation 1 unit up. The domain is {x|x ≥ 10-2stretch
Simplifying
Radical
Expressions
0}, and the range is {y|y ≤ 1}.
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.
Make a table of values.
x
0
f (x)
5
1
3
2
1
3
1
4
3
The domain is all real numbers, and the range is {y | y ≥ 0}.
69. h(x) = |x − 1|
SOLUTION: Since h(x) cannot be negative, the minimum point of the graph is where h(x) = 0.
Make a table of values.
x
–4
–2
h(x)
5
3
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0
1
2
1
4
3
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The domain is all real numbers, and the range is {y | y ≥ 0}.
10-2 Simplifying Radical Expressions
69. h(x) = |x − 1|
SOLUTION: Since h(x) cannot be negative, the minimum point of the graph is where h(x) = 0.
Make a table of values.
x
–4
–2
h(x)
5
3
0
1
2
1
4
3
The domain is all real numbers, and the range is {y | y ≥ 0}.
70. 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.
The domain is all real numbers, and the range is {y | y ≤ 1}.
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Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
2
71. x − 25 = 0
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domain isRadical
all real numbers,
and the range is {y | y ≥ 0}.
10-2The
Simplifying
Expressions
70. 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.
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.
2
71. x − 25 = 0
SOLUTION: For this equation, a = 1, b = 0, and c = –25.
The solutions are 5 and –5.
2
72. r + 25 = 0
SOLUTION: For this equation, a = 1, b = 0, and c = 25.
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10-2The
Simplifying
solutions Radical
are 5 andExpressions
–5.
2
72. r + 25 = 0
SOLUTION: For this equation, a = 1, b = 0, and c = 25.
There are no real positive square roots of –100. Therefore, the solution to this equation is ø.
2
73. 4w + 100 = 40w
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 4, b = –40, and c = 100.
The solution is 5.
2
74. 2r + r − 14 = 0
SOLUTION: For this equation, a = 2, b = 1, and c = –14.
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10-2 Simplifying Radical Expressions
The solution is 5.
2
74. 2r + r − 14 = 0
SOLUTION: For this equation, a = 2, b = 1, and c = –14.
The solutions are –2.9 and 2.4.
2
75. 5v − 7v = 1
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 5, b = –7, and c = –1.
The solutions are −0.1 and 1.5.
2
76. 11z − z = 3
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 11, b = –1, and c = –3.
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10-2 Simplifying Radical Expressions
The solutions are −0.1 and 1.5.
2
76. 11z − z = 3
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 11, b = –1, and c = –3.
The solutions are −0.5 and 0.6.
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime .
2
77. n − 81
SOLUTION: 78. 4 − 9a
2
SOLUTION: 5
79. 2x − 98x
3
SOLUTION: 4
80. 32x − 2y
4
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SOLUTION: 10-2 Simplifying Radical Expressions
4
80. 32x − 2y
4
SOLUTION: 2
81. 4t − 27
SOLUTION: In this trinomial, a = 4, b = 0 and c = –27, so m + p is zero and mp is negative. Therefore, m and p must be opposite
signs. List the factors of 4(–27) or –108 with a sum of 0.
Factors of –108
Sum of 0
1, –108
–107
107
–1, 108
2, –52
–50
50
–2, 52
3, –36
–33
33
–3, 36
4, –27
–23
23
–4, 27
6, –18
–12
12
–6, 18
9, –12
–3
3
–9, 12
2
There are no factors of –108 with a sum of 0, so 4t − 27 is prime.
3
2
82. x − 3x − 9x + 27
SOLUTION: 83. POPULATION The country of Latvia has been experiencing a 1.1% annual decrease in population. In 2009, its
population was 2,261,294. If the trend continues, predict Latvia’s population in 2019.
SOLUTION: Use the equation for exponential decay, with a = 2,261,294, r = 0.011, and t =10.
Latvia’s population in 2019 will be about 2,024,510 people.
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There
are more
84. TOMATOES
than 10,000 varieties of tomatoes. One seed company produces seed packages Page
for 37
200 varieties of tomatoes. For how many varieties do they not provide seeds?
SOLUTION: 10-2 Simplifying Radical Expressions
Latvia’s population in 2019 will be about 2,024,510 people.
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.
85. 24
SOLUTION: 86. 88
SOLUTION: 87. 180
SOLUTION: 88. 31
SOLUTION: The number 31 is prime. So, the prime factorization of 31 is 31.
89. 60
SOLUTION: 90. 90
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