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4-6 Square Roots and Cube Roots, Exams of Elementary Mathematics

Estimate each square root to the nearest ... The square root of 27 is between the integers 5 and ... Since 6 • 6 • 6 = 216, the cube root of 216 is 6.

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2021/2022

Uploaded on 09/12/2022

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Find each square root.
1.
SOLUTION:
= 4
ANSWER:
4
2.
SOLUTION:
= 10
ANSWER:
10
3.
SOLUTION:
= ±9
ANSWER:
±9
Estimate each square root to the nearest
integer.
4.
SOLUTION:
The first perfect square less than 27 is 25. =5
The first perfect square greater than 27 is 36. =
6
The square root of 27 is between the integers 5 and
6. Since 27 is closer to 25 than to 36, iscloser
to 5 than to 6.
ANSWER:
5
5.
SOLUTION:
The first perfect square less than 48 is 36. =6
The first perfect square greater than 48 is 49. =
7
The negative square root of 48 is between the
integers 6 and 7. Since 48 is closer to 49 than to
36, is closer to 7 than to 6.
ANSWER:
7
6.
SOLUTION:
The first perfect square less than 39 is 36.
=6
The first perfect square greater than 39 is 49. =
7
The positive and negative square roots of 39 are
between the integers ±6 and ±7. Since 39 is closer to
36 than to 49, ± is closer to ±6 than to ±7.
ANSWER:
±6
7.A baseball diamond is actually a square with an area
of 8100 square feet. Most baseball teams cover their
diamond with a tarp to protect it from the rain. The
sides are all the same length. How long is the tarp on
each side?
SOLUTION:
The length of each side of the tarp is equal to the
square root of the area.
The tarp is 90 feet long on each side.
ANSWER:
90 ft
Find each cube root.
8.
SOLUTION:
Since 8 •8 •8 = 512, the cube root of 512 is 8.
ANSWER:
8
9.
SOLUTION:
Since 13 •13 •13 = 2197, the cube root of 2197 is
13.
ANSWER:
13
10.
SOLUTION:
Since 10 10 10 = 1000, the cube root of
1000 is 10.
ANSWER:
10
11.
SOLUTION:
Since 7 7 7 = 343, the cube root of 343 is
7.
ANSWER:
7
Estimate each cube root to the nearest integer.
12.
SOLUTION:
The first perfect cube less than 74 is 64.
The first perfect cube greater than 74 is 125.
Approximate the placement of onanumber
line relative to 4 and 5. Since 74 is closer to 64 than
to 125, willbecloserto4.
.
ANSWER:
4
13.
SOLUTION:
The first perfect cube less than 39 is 27.
The first perfect cube greater than 39 is 64.
Approximate the placement of onanumber
line relative to 3 and 4. Since 39 is closer to 27 than
to 64, willbecloserto3.
ANSWER:
3
14.
SOLUTION:
The first perfect cube less than 636 is 729.
The first perfect cube greater than 636 is 512.
Approximate the placement of onanumber
line relative to 9 and 8. Since 636 is closer to
729 than to 512, willbecloserto9.
ANSWER:
9
15.
SOLUTION:
The first perfect cube less than 879 is 1000.
The first perfect cube greater than 879 is 729.
Approximate the placement of onanumber
line relative to 10 and 9. Since 879 is closer to
1000 than to 729, willbecloserto10.
ANSWER:
10
Find each square root.
16.
SOLUTION:
= 6
ANSWER:
6
17.
SOLUTION:
= 3
ANSWER:
3
18.
SOLUTION:
= 13
ANSWER:
13
19.
SOLUTION:
= 12
ANSWER:
12
20.
SOLUTION:
There is no real square root because no number
times itself is equal to 25.
ANSWER:
no real solution
21.
SOLUTION:
= ±1
ANSWER:
±1
Estimate each square root to the nearest
integer.
22.
SOLUTION:
The first perfect square less than 83 is 81. =9
The first perfect square greater than 83 is 100.
=10
The square root of 83 is between the integers 9 and
10. Since 83 is closer to 81 than to 100, iscloser
to 9 than to 10.
ANSWER:
9
23.
SOLUTION:
The first perfect square less than 34 is 25. =5
The first perfect square greater than 34 is 36. =
6
The square root of 34 is between the integers 5 and
6. Since 34 is closer to 36 than to 25, iscloser
to 6 than to 5.
ANSWER:
6
24.
SOLUTION:
The first perfect square less than 102 is 100. =
10
The first perfect square greater than 102 is 121.
=11
The negative square root of 102 is between the
integers 10 and 11. Since 102 is closer to 100 than
to 121, iscloserto10 than to 11.
ANSWER:
10
25.
SOLUTION:
The first perfect square less than 14 is 9. =3
The first perfect square greater than 14 is 16. =
4
The negative square root of 14 is between the
integers 3 and 4. Since 14 is closer to 16 than to 9,
iscloserto4 than to 3.
ANSWER:
4
26.
SOLUTION:
The first perfect square less than 78 is 64. =8
The first perfect square greater than 78 is 81. =
9
The positive and negative square roots of 78 are
between the integers ±8 and ±9. Since78 is closer to
81 than to 64, ± is closer to ±9 than to ±8.
ANSWER:
±9
27.
SOLUTION:
The first perfect square less than 146 is 144. =
12
The first perfect square greater than 146 is 169.
=13
The positive and negative square roots of 146 are
between the integers ±12 and ±13. Since 146 is
closer to 144 than to 169, ± is closer to ±12
than to ±13.
ANSWER:
±12
28.The table shows the heights of the tallest roller
coasters at Cedar Point. Use the formula from
Example 3 to determine how far a rider can see
from the highest point of each ride. Round to the
nearest tenth.
a.MillenniumForce
b.MeanStreak
c.HowmuchfarthercanariderseeontheTop
Thrill Dragster than on the Magnum XL-200?
SOLUTION:
a.
A rider can see about 21.5 miles from the highest
point on the Millennium Force roller coaster.
b.
A rider can see about 15.5 miles from the highest
point on the Mean Streak roller coaster.
c.
Top Thrill Dragster:
Magnum XL-200:
A rider can see 25.0 −17.5 or 7.5 miles farther on
the Top Thrill Dragster than on the Magnum XL-200.
ANSWER:
a.21.5mi
b.15.5mi
c.7.5mi
Find each cube root.
29.
SOLUTION:
Since 12 12 12 = 1728, the cube root of
1728 is 12.
ANSWER:
12
30.
SOLUTION:
Since 14 14 14 = 2744, the cube root of
2744 is 14.
ANSWER:
14
31.
SOLUTION:
Since 6 •6 •6 = 216, the cube root of 216 is 6.
ANSWER:
6
32.
SOLUTION:
Since 11 •11 •11 = 1331, the cube root of 1331 is
11.
ANSWER:
11
Estimate each cube root to the nearest integer.
Do not use a calculator.
33.
SOLUTION:
The first perfect cube less than 499 is 343.
The first perfect cube greater than 499 is 512.
Approximate the placement of onanumber
line relative to 7 and 8. Since 499 is closer to 512
than to 343, willbecloserto8.
ANSWER:
8
34.
SOLUTION:
The first perfect cube less than 576 is 512.
The first perfect cube greater than 576 is 729.
Approximate the placement of onanumber
line relative to 8 and 9. Since 576 is closer to 512
than to 729, willbecloserto8.
ANSWER:
8
35.
SOLUTION:
The first perfect cube less than 79 is 125.
The first perfect cube greater than 79 is 64.
Approximate the placement of onanumber
line relative to 5 and 4. Since 79 is closer to 64
than to 125, willbecloserto4.
ANSWER:
4
36.
SOLUTION:
The first perfect cube less than 1735 is 2197.
The first perfect cube greater than 1735 is 1728.
Approximate the placement of ona
number line relative to 13 and 12. Since 1735 is
closer to 1728 than to 2197, willbe
closer to 12.
ANSWER:
12
37.The area of a square is 215 square centimeters. Find
the length of a side to the nearest tenth. Then find its
approximate perimeter.
SOLUTION:
The length of each side of the square is equal to the
square root of its area.
A side length of the square is about 14.7 centimeters.
Use the formula to find the perimeter of the
square.
The approximate perimeter of the square is 58.8
centimeters.
ANSWER:
14.7 cm; 58.8 cm
38.Order , 8, , 9, 10, , from
least to greatest.
SOLUTION:
Write the given integers as square roots, and then
order the numbers from least to greatest.
8 =
9 =
10 =
< , < < < < <
So, the correct order is 10, , , 8,
, , 9.
ANSWER:
10, , , 8, , , 9
39.Reason Abstractly Write a number that completes
the analogy. x2 is to 121 as x3 is to ?.
SOLUTION:
121 is a perfect square of 11 since 11 •11 = 121.
1331 is a perfect cube of 11 since 11 •11 •11 =
1331.
So,x2isto121asx3 is to 1331.
ANSWER:
1331
40.Identify Structure Find a square root that lies
between 17 and 18.
SOLUTION:
and . Any number between 289
and 324 will have a square root between 17 and 18.
For example, .
ANSWER:
Sample answer:
41.PerseverewithProblemsUse inverse operations
to evaluate the following.
a.
b.
c.
SOLUTION:
a.
b.
c.
ANSWER:
a. 246
b.811
c.732
42.Building on the Essential Question Describe the
difference between an exact value and an
approximation when finding square roots of numbers
that are not perfect squares. Give an example of
each.
SOLUTION:
Sample answer: The exact value of a square root is
given using the square root symbol, such as . An
approximation is a decimal value, such as ≈6.
ANSWER:
Sample answer: The exact value of a square root is
given using the square root symbol, such as .
An approximation is a decimal value, such as
≈6.
43.Which point on the number line best represents
?
AA
BB
CC
DD
SOLUTION:
Use a calculator to estimate the square root of 210.
Point C best represents .
Choice C is the correct answer.
ANSWER:
C
44.Short Response Estimate the cube root of 65 to the
nearest integer.
SOLUTION:
The first perfect cube less than 65 is 64.
The first perfect cube greater than 65 is 125.
Approximate the placement of onanumber
line relative to 4 and 5. Since 65 is closer to 64 than
to 125, willbecloserto4.
ANSWER:
4
45.The new gymnasium at Oakdale Middle School has a
hardwood floor in the shape of a square. If the area
of the floor is 62,500 square feet, what is the length
of one side of the square floor?
F200ft
G225ft
H250ft
J275ft
SOLUTION:
The length of a side of the gymnasium floor is equal
to the square root of its area.
Each side has length 250 units.
Choice H is the correct answer.
ANSWER:
H
46.A surveyor determined the distance across a field
was feet.Whatistheapproximatedistance?
A25.6ft
B30.6ft
C39.6ft
D42.6ft
SOLUTION:
Use a calculator to estimate the square root of 1568
to the nearest tenth.
Choice C is the correct answer.
ANSWER:
C
Solve.
47.Joseph bought four books at a book sale. Each book
cost $4.50. He paid with $20.00. How much change
did he receive?
SOLUTION:
To find the cost of four books, multiply the number of
books by the cost for each book.
4×4.50=18.00
So, the four books cost $18.00.
To find the amount of change he received, subtract
the cost of the books from the amount that Joseph
paid.
20 18 = 2
So, Joseph received $2.00 in change.
ANSWER:
$2.00
48.Mrs. Tanner paid $18.00 for six boxes of pencils.
How much did each box of pencils cost?
SOLUTION:
To find the cost of each box of pencils, divide the
total cost by the number of boxes of pencils.
18÷6=3
So, each box of pencils cost $3.00.
ANSWER:
$3.00
49.Kathryn had $367.50 in her bank account. She wrote
a check for $25.00, and then withdrew $50.00 in
cash. She made a deposit of $100.00. How much
money is in her bank account now?
SOLUTION:
Kathryn has $392.50 in her bank account now.
Kathrynhad$367.50
in her bank account. $367.50
Shewroteacheckfor
$25.00. $367.50 $25.00 = $342.50
Shewithdrew$50.00
in cash. $342.50 $50.00 = $292.50
Shemadeadepositof
$100.00. $292.50+$100.00=$392.50
ANSWER:
$392.50
Name the property shown by each statement.
50.3 + 6 = 6 + 3
SOLUTION:
The order of the numbers changed. This is the
Commutative Property of Addition.
ANSWER:
Commutative Property (+)
51.13 + 0 = 13
SOLUTION:
When zero is added to any number, the sum is the
number. This is the Additive Identity Property.
ANSWER:
Additive Identity Property
52.(2 •5) + 6 = 6 + (2 •5)
SOLUTION:
The order of the numbers changed. This is the
Commutative Property of Addition.
ANSWER:
Commutative Property (+)
53.(x + 3) + 9 = x + (3 + 9)
SOLUTION:
The grouping of the numbers changed. This is the
Associative Property of Addition.
ANSWER:
Associative Property (+)
54.(y •2) •3 = y •(2 •3)
SOLUTION:
The grouping of the numbers changed. This is the
Associative Property of Multiplication.
ANSWER:
AssociativeProperty(×)
55.28 •1 = 28
SOLUTION:
When 1 is multiplied by any number, the product is
the number. This is the Multiplicative Identity
Property.
ANSWER:
Multiplicative Identity Property
56.n + t = t + n
SOLUTION:
The order of the variables has changed. This is the
Commutative Property of Addition.
ANSWER:
Commutative Property (+)
57.1652 •0 = 0
SOLUTION:
When zero is multiplied by any number, the product is
zero. This is the Zero Property of Multiplication.
ANSWER:
Zero Property of Multiplication
58.Suppose that four-tenths of the rectangle below is
shaded.
a. What fraction of the rectangle is not shaded?
b. What fraction of the rectangle would still need to
be shaded for half of the rectangle to be shaded?
c. If an additional of the original rectangle were
to be shaded, what fraction of the rectangle would be
shaded?
SOLUTION:
a. To find the fraction of the rectangle that is not
shaded, subtract the fraction that is shaded from one
whole.
of the rectangle is not shaded.
b. To find the fraction of the rectangle that would still
need to be shaded, subtract the fraction that is
already shaded from one-half.
of the rectangle would still need to be shaded for
half of the rectangle to be shaded.
c.
If an additional of the original rectangle were
shaded, of the rectangle would be shaded.
ANSWER:
a.
b.
c.
Find each product or quotient.
59.8 •(8)
SOLUTION:
The product of two integers with different signs is
negative. So, 8 •(8) = 64.
ANSWER:
64
60.4 •(12)
SOLUTION:
The product of two integers with the same sign is
positive. So, 4 •(12) = 48.
ANSWER:
48
61.40÷(5)
SOLUTION:
The quotient of two integers with different signs is
negative.So,40÷(5) = 8.
ANSWER:
8
62.150÷(25)
SOLUTION:
The quotient of two integers with the same sign is
positive. So, 150÷(25) = 6.
ANSWER:
6
Find each square root.
1.
SOLUTION:
= 4
ANSWER:
4
2.
SOLUTION:
= 10
ANSWER:
10
3.
SOLUTION:
= ±9
ANSWER:
±9
Estimate each square root to the nearest
integer.
4.
SOLUTION:
The first perfect square less than 27 is 25. =5
The first perfect square greater than 27 is 36. =
6
The square root of 27 is between the integers 5 and
6. Since 27 is closer to 25 than to 36, iscloser
to 5 than to 6.
ANSWER:
5
5.
SOLUTION:
The first perfect square less than 48 is 36. =6
The first perfect square greater than 48 is 49. =
7
The negative square root of 48 is between the
integers 6 and 7. Since 48 is closer to 49 than to
36, is closer to 7 than to 6.
ANSWER:
7
6.
SOLUTION:
The first perfect square less than 39 is 36.
=6
The first perfect square greater than 39 is 49. =
7
The positive and negative square roots of 39 are
between the integers ±6 and ±7. Since 39 is closer to
36 than to 49, ± is closer to ±6 than to ±7.
ANSWER:
±6
7.A baseball diamond is actually a square with an area
of 8100 square feet. Most baseball teams cover their
diamond with a tarp to protect it from the rain. The
sides are all the same length. How long is the tarp on
each side?
SOLUTION:
The length of each side of the tarp is equal to the
square root of the area.
The tarp is 90 feet long on each side.
ANSWER:
90 ft
Find each cube root.
8.
SOLUTION:
Since 8 •8 •8 = 512, the cube root of 512 is 8.
ANSWER:
8
9.
SOLUTION:
Since 13 •13 •13 = 2197, the cube root of 2197 is
13.
ANSWER:
13
10.
SOLUTION:
Since 10 10 10 = 1000, the cube root of
1000 is 10.
ANSWER:
10
11.
SOLUTION:
Since 7 7 7 = 343, the cube root of 343 is
7.
ANSWER:
7
Estimate each cube root to the nearest integer.
12.
SOLUTION:
The first perfect cube less than 74 is 64.
The first perfect cube greater than 74 is 125.
Approximate the placement of onanumber
line relative to 4 and 5. Since 74 is closer to 64 than
to 125, willbecloserto4.
.
ANSWER:
4
13.
SOLUTION:
The first perfect cube less than 39 is 27.
The first perfect cube greater than 39 is 64.
Approximate the placement of onanumber
line relative to 3 and 4. Since 39 is closer to 27 than
to 64, willbecloserto3.
ANSWER:
3
14.
SOLUTION:
The first perfect cube less than 636 is 729.
The first perfect cube greater than 636 is 512.
Approximate the placement of onanumber
line relative to 9 and 8. Since 636 is closer to
729 than to 512, willbecloserto9.
ANSWER:
9
15.
SOLUTION:
The first perfect cube less than 879 is 1000.
The first perfect cube greater than 879 is 729.
Approximate the placement of onanumber
line relative to 10 and 9. Since 879 is closer to
1000 than to 729, willbecloserto10.
ANSWER:
10
Find each square root.
16.
SOLUTION:
= 6
ANSWER:
6
17.
SOLUTION:
= 3
ANSWER:
3
18.
SOLUTION:
= 13
ANSWER:
13
19.
SOLUTION:
= 12
ANSWER:
12
20.
SOLUTION:
There is no real square root because no number
times itself is equal to 25.
ANSWER:
no real solution
21.
SOLUTION:
= ±1
ANSWER:
±1
Estimate each square root to the nearest
integer.
22.
SOLUTION:
The first perfect square less than 83 is 81. =9
The first perfect square greater than 83 is 100.
=10
The square root of 83 is between the integers 9 and
10. Since 83 is closer to 81 than to 100, iscloser
to 9 than to 10.
ANSWER:
9
23.
SOLUTION:
The first perfect square less than 34 is 25. =5
The first perfect square greater than 34 is 36. =
6
The square root of 34 is between the integers 5 and
6. Since 34 is closer to 36 than to 25, iscloser
to 6 than to 5.
ANSWER:
6
24.
SOLUTION:
The first perfect square less than 102 is 100. =
10
The first perfect square greater than 102 is 121.
=11
The negative square root of 102 is between the
integers 10 and 11. Since 102 is closer to 100 than
to 121, iscloserto10 than to 11.
ANSWER:
10
25.
SOLUTION:
The first perfect square less than 14 is 9. =3
The first perfect square greater than 14 is 16. =
4
The negative square root of 14 is between the
integers 3 and 4. Since 14 is closer to 16 than to 9,
iscloserto4 than to 3.
ANSWER:
4
26.
SOLUTION:
The first perfect square less than 78 is 64. =8
The first perfect square greater than 78 is 81. =
9
The positive and negative square roots of 78 are
between the integers ±8 and ±9. Since78 is closer to
81 than to 64, ± is closer to ±9 than to ±8.
ANSWER:
±9
27.
SOLUTION:
The first perfect square less than 146 is 144. =
12
The first perfect square greater than 146 is 169.
=13
The positive and negative square roots of 146 are
between the integers ±12 and ±13. Since 146 is
closer to 144 than to 169, ± is closer to ±12
than to ±13.
ANSWER:
±12
28.The table shows the heights of the tallest roller
coasters at Cedar Point. Use the formula from
Example 3 to determine how far a rider can see
from the highest point of each ride. Round to the
nearest tenth.
a.MillenniumForce
b.MeanStreak
c.HowmuchfarthercanariderseeontheTop
Thrill Dragster than on the Magnum XL-200?
SOLUTION:
a.
A rider can see about 21.5 miles from the highest
point on the Millennium Force roller coaster.
b.
A rider can see about 15.5 miles from the highest
point on the Mean Streak roller coaster.
c.
Top Thrill Dragster:
Magnum XL-200:
A rider can see 25.0 −17.5 or 7.5 miles farther on
the Top Thrill Dragster than on the Magnum XL-200.
ANSWER:
a.21.5mi
b.15.5mi
c.7.5mi
Find each cube root.
29.
SOLUTION:
Since 12 12 12 = 1728, the cube root of
1728 is 12.
ANSWER:
12
30.
SOLUTION:
Since 14 14 14 = 2744, the cube root of
2744 is 14.
ANSWER:
14
31.
SOLUTION:
Since 6 •6 •6 = 216, the cube root of 216 is 6.
ANSWER:
6
32.
SOLUTION:
Since 11 •11 •11 = 1331, the cube root of 1331 is
11.
ANSWER:
11
Estimate each cube root to the nearest integer.
Do not use a calculator.
33.
SOLUTION:
The first perfect cube less than 499 is 343.
The first perfect cube greater than 499 is 512.
Approximate the placement of onanumber
line relative to 7 and 8. Since 499 is closer to 512
than to 343, willbecloserto8.
ANSWER:
8
34.
SOLUTION:
The first perfect cube less than 576 is 512.
The first perfect cube greater than 576 is 729.
Approximate the placement of onanumber
line relative to 8 and 9. Since 576 is closer to 512
than to 729, willbecloserto8.
ANSWER:
8
35.
SOLUTION:
The first perfect cube less than 79 is 125.
The first perfect cube greater than 79 is 64.
Approximate the placement of onanumber
line relative to 5 and 4. Since 79 is closer to 64
than to 125, willbecloserto4.
ANSWER:
4
36.
SOLUTION:
The first perfect cube less than 1735 is 2197.
The first perfect cube greater than 1735 is 1728.
Approximate the placement of ona
number line relative to 13 and 12. Since 1735 is
closer to 1728 than to 2197, willbe
closer to 12.
ANSWER:
12
37.The area of a square is 215 square centimeters. Find
the length of a side to the nearest tenth. Then find its
approximate perimeter.
SOLUTION:
The length of each side of the square is equal to the
square root of its area.
A side length of the square is about 14.7 centimeters.
Use the formula to find the perimeter of the
square.
The approximate perimeter of the square is 58.8
centimeters.
ANSWER:
14.7 cm; 58.8 cm
38.Order , 8, , 9, 10, , from
least to greatest.
SOLUTION:
Write the given integers as square roots, and then
order the numbers from least to greatest.
8 =
9 =
10 =
< , < < < < <
So, the correct order is 10, , , 8,
, , 9.
ANSWER:
10, , , 8, , , 9
39.Reason Abstractly Write a number that completes
the analogy. x2 is to 121 as x3 is to ?.
SOLUTION:
121 is a perfect square of 11 since 11 •11 = 121.
1331 is a perfect cube of 11 since 11 •11 •11 =
1331.
So,x2isto121asx3 is to 1331.
ANSWER:
1331
40.Identify Structure Find a square root that lies
between 17 and 18.
SOLUTION:
and . Any number between 289
and 324 will have a square root between 17 and 18.
For example, .
ANSWER:
Sample answer:
41.PerseverewithProblemsUse inverse operations
to evaluate the following.
a.
b.
c.
SOLUTION:
a.
b.
c.
ANSWER:
a. 246
b.811
c.732
42.Building on the Essential Question Describe the
difference between an exact value and an
approximation when finding square roots of numbers
that are not perfect squares. Give an example of
each.
SOLUTION:
Sample answer: The exact value of a square root is
given using the square root symbol, such as . An
approximation is a decimal value, such as ≈6.
ANSWER:
Sample answer: The exact value of a square root is
given using the square root symbol, such as .
An approximation is a decimal value, such as
≈6.
43.Which point on the number line best represents
?
AA
BB
CC
DD
SOLUTION:
Use a calculator to estimate the square root of 210.
Point C best represents .
Choice C is the correct answer.
ANSWER:
C
44.Short Response Estimate the cube root of 65 to the
nearest integer.
SOLUTION:
The first perfect cube less than 65 is 64.
The first perfect cube greater than 65 is 125.
Approximate the placement of onanumber
line relative to 4 and 5. Since 65 is closer to 64 than
to 125, willbecloserto4.
ANSWER:
4
45.The new gymnasium at Oakdale Middle School has a
hardwood floor in the shape of a square. If the area
of the floor is 62,500 square feet, what is the length
of one side of the square floor?
F200ft
G225ft
H250ft
J275ft
SOLUTION:
The length of a side of the gymnasium floor is equal
to the square root of its area.
Each side has length 250 units.
Choice H is the correct answer.
ANSWER:
H
46.A surveyor determined the distance across a field
was feet.Whatistheapproximatedistance?
A25.6ft
B30.6ft
C39.6ft
D42.6ft
SOLUTION:
Use a calculator to estimate the square root of 1568
to the nearest tenth.
Choice C is the correct answer.
ANSWER:
C
Solve.
47.Joseph bought four books at a book sale. Each book
cost $4.50. He paid with $20.00. How much change
did he receive?
SOLUTION:
To find the cost of four books, multiply the number of
books by the cost for each book.
4×4.50=18.00
So, the four books cost $18.00.
To find the amount of change he received, subtract
the cost of the books from the amount that Joseph
paid.
20 18 = 2
So, Joseph received $2.00 in change.
ANSWER:
$2.00
48.Mrs. Tanner paid $18.00 for six boxes of pencils.
How much did each box of pencils cost?
SOLUTION:
To find the cost of each box of pencils, divide the
total cost by the number of boxes of pencils.
18÷6=3
So, each box of pencils cost $3.00.
ANSWER:
$3.00
49.Kathryn had $367.50 in her bank account. She wrote
a check for $25.00, and then withdrew $50.00 in
cash. She made a deposit of $100.00. How much
money is in her bank account now?
SOLUTION:
Kathryn has $392.50 in her bank account now.
Kathrynhad$367.50
in her bank account. $367.50
Shewroteacheckfor
$25.00. $367.50 $25.00 = $342.50
Shewithdrew$50.00
in cash. $342.50 $50.00 = $292.50
Shemadeadepositof
$100.00. $292.50+$100.00=$392.50
ANSWER:
$392.50
Name the property shown by each statement.
50.3 + 6 = 6 + 3
SOLUTION:
The order of the numbers changed. This is the
Commutative Property of Addition.
ANSWER:
Commutative Property (+)
51.13 + 0 = 13
SOLUTION:
When zero is added to any number, the sum is the
number. This is the Additive Identity Property.
ANSWER:
Additive Identity Property
52.(2 •5) + 6 = 6 + (2 •5)
SOLUTION:
The order of the numbers changed. This is the
Commutative Property of Addition.
ANSWER:
Commutative Property (+)
53.(x + 3) + 9 = x + (3 + 9)
SOLUTION:
The grouping of the numbers changed. This is the
Associative Property of Addition.
ANSWER:
Associative Property (+)
54.(y •2) •3 = y •(2 •3)
SOLUTION:
The grouping of the numbers changed. This is the
Associative Property of Multiplication.
ANSWER:
AssociativeProperty(×)
55.28 •1 = 28
SOLUTION:
When 1 is multiplied by any number, the product is
the number. This is the Multiplicative Identity
Property.
ANSWER:
Multiplicative Identity Property
56.n + t = t + n
SOLUTION:
The order of the variables has changed. This is the
Commutative Property of Addition.
ANSWER:
Commutative Property (+)
57.1652 •0 = 0
SOLUTION:
When zero is multiplied by any number, the product is
zero. This is the Zero Property of Multiplication.
ANSWER:
Zero Property of Multiplication
58.Suppose that four-tenths of the rectangle below is
shaded.
a. What fraction of the rectangle is not shaded?
b. What fraction of the rectangle would still need to
be shaded for half of the rectangle to be shaded?
c. If an additional of the original rectangle were
to be shaded, what fraction of the rectangle would be
shaded?
SOLUTION:
a. To find the fraction of the rectangle that is not
shaded, subtract the fraction that is shaded from one
whole.
of the rectangle is not shaded.
b. To find the fraction of the rectangle that would still
need to be shaded, subtract the fraction that is
already shaded from one-half.
of the rectangle would still need to be shaded for
half of the rectangle to be shaded.
c.
If an additional of the original rectangle were
shaded, of the rectangle would be shaded.
ANSWER:
a.
b.
c.
Find each product or quotient.
59.8 •(8)
SOLUTION:
The product of two integers with different signs is
negative. So, 8 •(8) = 64.
ANSWER:
64
60.4 •(12)
SOLUTION:
The product of two integers with the same sign is
positive. So, 4 •(12) = 48.
ANSWER:
48
61.40÷(5)
SOLUTION:
The quotient of two integers with different signs is
negative.So,40÷(5) = 8.
ANSWER:
8
62.150÷(25)
SOLUTION:
The quotient of two integers with the same sign is
positive. So, 150÷(25) = 6.
ANSWER:
6
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Find each square root.

SOLUTION: = 4 ANSWER: 4

SOLUTION: = − 10 ANSWER: − 10

SOLUTION: = ± 9 ANSWER: ± 9 Estimate each square root to the nearest integer.

SOLUTION: The first perfect square less than 27 is 25. = 5 The first perfect square greater than 27 is 36. = 6 The square root of 27 is between the integers 5 and

  1. Since 27 is closer to 25 than to 36, is closer to 5 than to 6. ANSWER: 5

SOLUTION: The first perfect square less than 48 is 36. = 6 The first perfect square greater than 48 is 49. = 7 The negative square root of 48 is between the integers −6 and −7. Since 48 is closer to 49 than to 36, − is closer to −7 than to −6. ANSWER: − 7

  1. Since 27 is closer to 25 than to 36, is closer to 5 than to 6. ANSWER: 5

SOLUTION: The first perfect square less than 48 is 36. = 6 The first perfect square greater than 48 is 49. = 7 The negative square root of 48 is between the integers −6 and −7. Since 48 is closer to 49 than to 36, − is closer to −7 than to −6. ANSWER: − 7

SOLUTION: The first perfect square less than 39 is 36. = 6 The first perfect square greater than 39 is 49. = 7 The positive and negative square roots of 39 are between the integers ±6 and ±7. Since 39 is closer to 36 than to 49, ± is closer to ± 6 than to ± 7. ANSWER: ± 6

  1. A baseball diamond is actually a square with an area of 8100 square feet. Most baseball teams cover their diamond with a tarp to protect it from the rain. The sides are all the same length. How long is the tarp on each side? SOLUTION: The length of each side of the tarp is equal to the square root of the area. The tarp is 90 feet long on each side. ANSWER: 90 ft Find each cube root.

SOLUTION: Since 8 • 8 • 8 = 512, the cube root of 512 is 8. ANSWER: 8 eSolutions Manual - Powered by Cognero Page 1

4 - 6 Square Roots and Cube Roots

The tarp is 90 feet long on each side. ANSWER: 90 ft Find each cube root.

SOLUTION: Since 8 • 8 • 8 = 512, the cube root of 512 is 8. ANSWER: 8

SOLUTION: Since 13 • 13 • 13 = 2197, the cube root of 2197 is

ANSWER: 13

SOLUTION: Since − 10 • − 10 • − 10 = −1000, the cube root of − 1000 is −10. ANSWER: − 10

SOLUTION: Since − 7 • − 7 • − 7 = −343, the cube root of − 343 is −7. ANSWER: − 7 Estimate each cube root to the nearest integer.

SOLUTION: The first perfect cube less than 74 is 64. The first perfect cube greater than 74 is 125. Approximate the placement of on a number line relative to 4 and 5. Since 74 is closer to 64 than to 125, will be closer to 4. .

ANSWER:

Estimate each cube root to the nearest integer.

SOLUTION: The first perfect cube less than 74 is 64. The first perfect cube greater than 74 is 125. Approximate the placement of on a number line relative to 4 and 5. Since 74 is closer to 64 than to 125, will be closer to 4. . ANSWER: 4

SOLUTION: The first perfect cube less than 39 is 27. The first perfect cube greater than 39 is 64. Approximate the placement of on a number line relative to 3 and 4. Since 39 is closer to 27 than to 64, will be closer to 3. ANSWER: 3

SOLUTION: The first perfect cube less than – 636 is – 729. The first perfect cube greater than – 636 is – 512. Approximate the placement of on a number line relative to – 9 and – 8. Since – 636 is closer to – 729 than to – 512, will be closer to – 9. ANSWER: eSolutions Manual - Powered by Cognero Page 2

4 - 6 Square Roots and Cube Roots

SOLUTION:

ANSWER:

Estimate each square root to the nearest integer.

SOLUTION: The first perfect square less than 83 is 81. = 9 The first perfect square greater than 83 is 100. = 10 The square root of 83 is between the integers 9 and

  1. Since 83 is closer to 81 than to 100, is closer to 9 than to 10. ANSWER: 9

SOLUTION: The first perfect square less than 34 is 25. = 5 The first perfect square greater than 34 is 36. = 6 The square root of 34 is between the integers 5 and

  1. Since 34 is closer to 36 than to 25, is closer to 6 than to 5. ANSWER: 6

SOLUTION: The first perfect square less than 102 is 100. = 10 The first perfect square greater than 102 is 121. = 11 The negative square root of 102 is between the integers – 10 and – 11. Since 102 is closer to 100 than to 121, is closer to – 10 than to – 11. ANSWER: − 10

SOLUTION: The first perfect square less than 14 is 9. = 3 The first perfect square greater than 14 is 16. = 4 The negative square root of 14 is between the integers – 3 and – 4. Since 14 is closer to 16 than to 9, is closer to – 4 than to – 3. integers – 10 and – 11. Since 102 is closer to 100 than to 121, is closer to – 10 than to – 11. ANSWER: − 10

SOLUTION: The first perfect square less than 14 is 9. = 3 The first perfect square greater than 14 is 16. = 4 The negative square root of 14 is between the integers – 3 and – 4. Since 14 is closer to 16 than to 9, is closer to – 4 than to – 3. ANSWER: − 4

SOLUTION: The first perfect square less than 78 is 64. = 8 The first perfect square greater than 78 is 81. = 9 The positive and negative square roots of 78 are between the integers ±8 and ±9. Since78 is closer to 81 than to 64, ± is closer to ± 9 than to ± 8. ANSWER: ± 9

SOLUTION: The first perfect square less than 146 is 144. = 12 The first perfect square greater than 146 is 169. = 13 The positive and negative square roots of 146 are between the integers ±12 and ±13. Since 146 is closer to 144 than to 169, ± is closer to ± 12 than to ± 13. ANSWER: ± 12

28. The table shows the heights of the tallest roller

coasters at Cedar Point. Use the formula from

Example 3 to determine how far a rider can see

from the highest point of each ride. Round to the

nearest tenth.

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4 - 6 Square Roots and Cube Roots

28. The table shows the heights of the tallest roller

coasters at Cedar Point. Use the formula from

Example 3 to determine how far a rider can see

from the highest point of each ride. Round to the

nearest tenth.

a. (^) Millennium Force b. Mean Streak c. (^) How much farther can a rider see on the Top Thrill Dragster than on the Magnum XL-200? SOLUTION: a. A rider can see about 21.5 miles from the highest point on the Millennium Force roller coaster. b. A rider can see about 15.5 miles from the highest point on the Mean Streak roller coaster. c. Top Thrill Dragster: Magnum XL-200: A rider can see 25.0 − 17.5 or 7.5 miles farther on the Top Thrill Dragster than on the Magnum XL-200. ANSWER: a. (^) 21.5 mi b. (^) 15.5 mi c. (^) 7.5 mi Find each cube root.

ANSWER:

a. (^) 21.5 mi b. (^) 15.5 mi c. (^) 7.5 mi Find each cube root.

SOLUTION: Since − 12 • − 12 • − 12 = − 1728 , the cube root of − 1728 is − 12. ANSWER: − 12

SOLUTION: Since − 14 • − 14 • − 14 = − 2744 , the cube root of − 2744 is − 14. ANSWER: − 14

SOLUTION: Since 6 • 6 • 6 = 216, the cube root of 216 is 6. ANSWER: 6

SOLUTION: Since 11 • 11 • 11 = 1331, the cube root of 1331 is

ANSWER: 11 Estimate each cube root to the nearest integer. Do not use a calculator.

SOLUTION: The first perfect cube less than 499 is 343. The first perfect cube greater than 499 is 512. Approximate the placement of on a number line relative to 7 and 8. Since 499 is closer to 512 than to 343, will be closer to 8. eSolutions Manual - Powered by Cognero Page 5

4 - 6 Square Roots and Cube Roots

ANSWER:

  1. The area of a square is 215 square centimeters. Find the length of a side to the nearest tenth. Then find its approximate perimeter. SOLUTION: The length of each side of the square is equal to the square root of its area. A side length of the square is about 14.7 centimeters. Use the formula to find the perimeter of the square. The approximate perimeter of the square is 58. centimeters. ANSWER: 14.7 cm; 58.8 cm
  2. Order , −8, , 9, −10, , from least to greatest. SOLUTION: Write the given integers as square roots, and then order the numbers from least to greatest. −8 = 9 = −10 = < , < < < < < So, the correct order is −10, , , −8, , , 9. ANSWER: −10, , , −8, , , 9
  3. Reason Abstractly Write a number that completes the analogy. x 2 is to 121 as x 3 is to?. SOLUTION: 121 is a perfect square of 11 since 11 • 11 = 121. 1331 is a perfect cube of 11 since 11 • 11 • 11 =
    So, x 2 is to 121 as x 3 is to 1331. ANSWER: 1331
  4. Identify Structure^ Find a square root that lies between 17 and 18.

ANSWER:

  1. Reason Abstractly^ Write a number that completes the analogy. x 2 is to 121 as x 3 is to?. SOLUTION: 121 is a perfect square of 11 since 11 • 11 = 121. 1331 is a perfect cube of 11 since 11 • 11 • 11 =
    So, x^2 is to 121 as x^3 is to 1331. ANSWER: 1331
  2. Identify Structure^ Find a square root that lies between 17 and 18. SOLUTION: and. Any number between 289 and 324 will have a square root between 17 and 18. For example,. ANSWER: Sample answer:
  3. Persevere with Problems Use inverse operations to evaluate the following. a. b. c. SOLUTION: a. b. c. ANSWER: a. (^) 246 b. (^) 811 c. (^) 732
  4. Building on the Essential Question Describe the difference between an exact value and an approximation when finding square roots of numbers that are not perfect squares. Give an example of each. SOLUTION: Sample answer: The exact value of a square root is given using the square root symbol, such as. An approximation is a decimal value, such as ≈ 6. eSolutions Manual - Powered by Cognero Page 7

4 - 6 Square Roots and Cube Roots

ANSWER:

a. (^) 246 b. 811 c. (^) 732

  1. Building on the Essential Question^ Describe the difference between an exact value and an approximation when finding square roots of numbers that are not perfect squares. Give an example of each. SOLUTION: Sample answer: The exact value of a square root is given using the square root symbol, such as. An approximation is a decimal value, such as ≈ 6. ANSWER: Sample answer: The exact value of a square root is given using the square root symbol, such as. An approximation is a decimal value, such as ≈ 6.
  2. Which point on the number line best represents ?

A A

B B

C C

D D

SOLUTION:

Use a calculator to estimate the square root of 210. Point C best represents. Choice C is the correct answer. ANSWER: C

  1. Short Response^ Estimate the cube root of 65 to the nearest integer. SOLUTION: The first perfect cube less than 65 is 64. The first perfect cube greater than 65 is 125. Approximate the placement of on a number line relative to 4 and 5. Since 65 is closer to 64 than to 125, will be closer to 4. Point C best represents. Choice C is the correct answer. ANSWER: C
  2. Short Response^ Estimate the cube root of 65 to the nearest integer. SOLUTION: The first perfect cube less than 65 is 64. The first perfect cube greater than 65 is 125. Approximate the placement of on a number line relative to 4 and 5. Since 65 is closer to 64 than to 125, will be closer to 4. ANSWER: 4
  3. The new gymnasium at Oakdale Middle School has a hardwood floor in the shape of a square. If the area of the floor is 62,500 square feet, what is the length of one side of the square floor?

F 200 ft

G 225 ft

H 250 ft

J 275 ft

SOLUTION:

The length of a side of the gymnasium floor is equal to the square root of its area. Each side has length 250 units. Choice H is the correct answer. ANSWER: H

  1. A surveyor determined the distance across a field was feet. What is the approximate distance? A 25.6 ft B 30.6 ft C 39.6 ft D 42.6 ft SOLUTION: Use a calculator to estimate the square root of 1568 to the nearest tenth. Choice C is the correct answer. eSolutions Manual - Powered by Cognero Page 8

4 - 6 Square Roots and Cube Roots

The order of the numbers changed. This is the Commutative Property of Addition. ANSWER: Commutative Property (+)

  1. ( x + 3) + 9 = x + (3 + 9) SOLUTION: The grouping of the numbers changed. This is the Associative Property of Addition. ANSWER: Associative Property (+)
  2. ( y • 2) • 3 = y • ( 2 • 3) SOLUTION: The grouping of the numbers changed. This is the Associative Property of Multiplication. ANSWER: Associative Property (×)
  3. 28 • 1 = 28 SOLUTION: When 1 is multiplied by any number, the product is the number. This is the Multiplicative Identity Property. ANSWER: Multiplicative Identity Property
  4. n + t = t + n SOLUTION: The order of the variables has changed. This is the Commutative Property of Addition. ANSWER: Commutative Property (+)
  5. 1652 • 0 = 0 SOLUTION: When zero is multiplied by any number, the product is zero. This is the Zero Property of Multiplication. ANSWER: Zero Property of Multiplication
  6. Suppose that four-tenths of the rectangle below is shaded. a. (^) What fraction of the rectangle is not shaded? b. (^) What fraction of the rectangle would still need to be shaded for half of the rectangle to be shaded? c. (^) If an additional of the original rectangle were to be shaded, what fraction of the rectangle would be a. (^) What fraction of the rectangle is not shaded? b. What fraction of the rectangle would still need to be shaded for half of the rectangle to be shaded? c. (^) If an additional of the original rectangle were to be shaded, what fraction of the rectangle would be shaded? SOLUTION: a. To find the fraction of the rectangle that is not shaded, subtract the fraction that is shaded from one whole. of the rectangle is not shaded. b. (^) To find the fraction of the rectangle that would still need to be shaded, subtract the fraction that is already shaded from one-half. of the rectangle would still need to be shaded for half of the rectangle to be shaded. c. If an additional of the original rectangle were shaded, of the rectangle would be shaded. ANSWER: a. eSolutions Manual - Powered by Cognero Page 10

4 - 6 Square Roots and Cube Roots

If an additional of the original rectangle were shaded, of the rectangle would be shaded. ANSWER: a. b. c. Find each product or quotient.

  1. 8 • (–8) SOLUTION: The product of two integers with different signs is negative. So, 8 • (–8) = – 64. ANSWER:
    • 64
  2. – 4 • (–12) SOLUTION: The product of two integers with the same sign is positive. So, – 4 • (–12) = 48. ANSWER: 48
  3. 40 ÷ (–5) SOLUTION: The quotient of two integers with different signs is negative. So, 40 ÷ (–5) = – 8. ANSWER:
    • 8
  4. – 150 ÷ (–25) SOLUTION: The quotient of two integers with the same sign is positive. So, – 150 ÷ (–25) = 6. ANSWER: 6 eSolutions Manual - Powered by Cognero Page 11

4 - 6 Square Roots and Cube Roots