Download Numerical Roots and Radicals Chapter Questions

Numbers & Operations 8th Grade Chapter Questions
1.
What are the properties of a square?
2.
What does taking the square root have to do with the area of a square?
3.
Why is it helpful to memorize perfect squares?
4.
What can be helpful when finding the square roots of numbers greater than 400?
5.
Explain how to take the square root of a fraction or a decimal.
6.
Explain how to approximate a square root.
7.
What is the difference between an irrational and rational number?
8.
Explain how to convert between the different forms of a rational number.
Numbers & Operations Chapter Problems
Converting Repeating Decimals to Fractions and Decimal Expansion
Classwork
1. Write each repeating decimal as a fraction in simplest form *
a.
b.
c.
d.
e.
f.
2. Find the decimal expansion of the following*:
a.
b.
c.
d.
e.
f.
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Homework
3. Write each repeating decimal as a fraction in simplest form *
a.
b.
c.
d.
e.
f.
4. Find the decimal expansion of the following*:
a.
b.
c.
d.
e.
f.
Squares, Square Roots & Perfect Squares
Classwork
5. A square has an area of 9 units2.
a. What is the side length of a square of this area?
b. Draw a square with an area of 9 units2.
c.
What is the square root of 9?
d. Explain why your answers in parts (a) and (c) are the same.
*
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6. Fill in the following table:
Side Length
of a square
Area of
the square
(units)
(units2)
1
2
3
4
5
6
7
8
9
10
11
12
13
7. Explain how the table above helps you find the square root of 121?
8. Simplify each square root.
a.
b.
c.
d.
e.
Homework
9. A square has an area of 36 units2.
a. What is the side length of a square of this area?
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b. Draw a square with an area of 36 units2.
c.
What is the square root of 36?
d. Explain why your answers in parts (a) and (c) are the same.
10. Fill in the following table:
Side Length
of a square
Area of the
square
(units)
(units2)
14
15
16
17
18
19
20
11. Simplify each square root.
a.
b.
c.
d.
e.
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Squares of Numbers Greater Than 20
Classwork
12. Fill in the following table:
Side Length
of a square
Area of the
square
(units)
(units2)
10
20
30
40
50
60
70
80
90
100
13. If you compare that to the table of side lengths from 1-10, what pattern do you notice?
14. Simplify each square root.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
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Homework
15. Simplify each square root.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
Simplifying Perfect Square Radical Expressions
Classwork
16. Simplify each square root.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
Homework
17. Simplify each square root.
a.
b. c.
d.
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e.
f.
g.
h.
i.
-
j.
k.
l.
m.
n.
o.
Approximating Square Roots
Classwork
18. What two integers do the following square roots fall between?
a.
b.
c.
d.
e.
19. Draw and label a number line from 0 to 10. Place the following square roots on the number line.
a.
b.
c.
d.
e.
20. Estimate the following square roots.
a.
b.
c.
d.
e.
f.
g.
h.
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i.
j.
21. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
22. For what integer x is
closest to 7.42?*
23. For what integer x is
closest to 5.1?*
24. For what integer x is
closest to 3.9?*
Homework
25. What two integers do the following square roots fall between?
a.
b.
c.
d.
e.
26. Draw and label a number line from 0 to 10. Place the following square roots on the number line.
a.
b.
c.
d.
e.
27. Estimate the following square roots.
a.
b.
c.
d.
e.
f.
g.
*
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h.
i.
j.
28. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
29. For what integer x is
closest to 5.2?*
30. For what integer x is
closest to 6.1?*
31. For what integer x is
closest to 6.9?*
Rational & Irrational Numbers
Classwork.
32. Circle the numbers below that are rational
a. 3.5
b.
c.
π
d.
e.
f.
g.
h.
0.25
i.
j.
0.
33. Given the statement “If x is a rational number then
statement
false?*
A) 15 B) 24
C) 4 D) 20
34. Given the statement “If x is a rational number then
statement
*
false?*
A) 25 B) 49
is irrational.” Which values of x make the
is irrational.” Which values of x make the
C) 5 D) 59
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Homework
35. Circle the numbers below that are irrational.
a.
b.
c.
d.
e.
6.75
f.
g.
h. π
i.
j.
0.
36. Given the statement “If x is a rational number then
statement false?* A) 36 B) 8
C) 16 D) 121
37. Given the statement “If x is a rational number then
statement
false?*
A) 144 B) 12
is irrational.” Which values of x make the
is irrational.” Which values of x make the
C) 48 D)30
Properties of Exponents
Classwork
38. Complete each equation for the missing value:
a. (52)(55) = 5?
b. (127)(123) = 12?
c.
(3-2)(35) = 3?
d. (49)(4-3) = 4?
e. (54)(5?) = 512
f.
(107)(10?)(10-6) = 103
g. 34 ÷ 32 = 3?
*
h.
59
= 5?
56
i.
95
= 9?
98
j.
124 ÷ 126 = 12?
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k.
108 ÷ 10? = 103
l.
2?
= 24
23
39. A rectangle has a length of
rectangle as a power of
mm and a width of
mm. Write an expression for the area of the
5.*
inches as a power of 7.*
40. Express the volume of a cube with a side length of
41. a) Write an exponential expression for the area of a rectangle with a length of
of
meters. b) Evaluate the expression to find the area of the
meters and a width
rectangle.*
Homework
42. Complete each equation for the missing value:
a. (122)(127) = 12?
b. (25)(22) = 2?
c.
(5-3)(55) = 5?
d. (158)(15-5) = 15?
e. (67)(6?) = 615
f.
(11-6)(11?)(118) = 115
g. 77 ÷ 73 = 7?
1110
h.
= 11?
116
i.
37 ÷ 39 = 3?
j.
26
= 2?
210
k.
136
= 132
13?
l.
5? ÷ 56 = 53
43. A rectangle has a length of
rectangle as a power of
mm and a width of
mm. Write an expression for the area of the
4.*
44. Express the volume of a cube with a side length of
*
*
inches as a power of 2.*
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45. a) Write an exponential expression for the area of a rectangle with a length of
of
meters. b) Evaluate the expression to find the area of the
meters and a width
rectangle.*
Roots & Radicals Review
Determine whether the given numbers are perfect squares. Circle your answer.
1) 1
Yes
No
2) 8
Yes
No
3) 16
Yes
No
4) 25
Yes
No
5) 82
Yes
No
Circle the simplified version of each square root:
6)
a.
b.
c.
d.
14
12
72
21
7)
a. 10
b. 6
c. 0.6
d. 18
8)
a. -7
b. 0.7
c. 0.07
d. -0.07
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Circle whether the given number is rational or irrational
9) π
rational
irrational
10) 0.875
rational
irrational
11) )
rational
irrational
12) Between what two integers does the following square root fall?
a.
b.
c.
d.
4&5
6&7
7&8
5&6
13) (47)(43) = 4?
a. 10
b. 24
c. 4
d. 5
14) Approximate
________
15) Find the missing value 114 ÷ 116 = 11? ____________________
________________
16)
17) (67)(6-2) = 6?
18) A rectangle has a length of
cm and a width of
cm. What is the area of the rectangle written as a
power of 4? ___________________
19) Draw and label a number line from 0-10. Place the following numbers on the number line: *
,
*
,
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20) Write
21) Write
as a fraction in simplest form.*
as a fraction in simplest form.*
22) Write two exponential expressions with like bases. Leave all answers in simplified exponential
form.
a. Expression 1:
Expression 2:
b. Multiply your expressions.
c.
Divide your expressions.
d. Raise your first expression to the 5th power.
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Answer Key
5.
1.
a. 3 units
b.
a.
b.
c.
3 units
d.
e.
f.
3 units
7
2.
a.
b.
c.
d.
e.
f.
c. 3
d. Area of a Square = Side2 and 9 = 32
6.
Side Length
of a Square
Area of the
square
(units)
(units2)
a.
1
1
b.
2
4
c.
3
9
4
16
5
25
6
36
7
49
2x2 y3 z 2 6
8
64
x3 y 2 z 3
9
81
2 x y z 14 x
10
100
x2 y 2 z3 10 yz
11
121
g. 2 x 2 yz 2 7 yz
12
144
13
169
3.
d.
e.
f.
4
4.
2 4 2
a. 2 x y z 10 xz
b.
c.
d.
e.
f.
h.
i.
x 3 y 3 z 3 yz
65 y
x y 4 z 22 z
3x 4 y 3 z
3xy
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7.
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Since Area of a square = side2, the square
root of the area = side. So, √121 = 11.
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8.
12.
a.
b.
c.
d.
e.
5
8
9
7
4
9.
a. 6 units
b.
6 units
6 units
c. 6
d. Area = Side2 and 36 = 62
10.
Side Length
of a square
Area of the
square
(units)
(units2)
14
196
15
225
16
256
17
289
18
324
19
361
20
400
17
20
14
19
12
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Area of the
square
(units)
(units2)
10
100
20
400
30
900
40
1600
50
2500
60
3600
70
4900
80
6400
90
8100
100
10,000
13. Each answer in this table is 100 times
greater than the corresponding answer in
the other table. (or 102 times greater).
14.
a. 53
b. 89
c. 22
d. 80
e. 45
f. 15
g. 29
h. 97
i. 31
j. 66
15.
a. 71
b. 36
c. 92
d. 55
e. 61
f. 83
g. 24
h. 49
i. 50
11.
a.
b.
c.
d.
e.
Side Length
of a square
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j.
17
a.
b.
c.
d.
e.
5
8
-9
No real solution
7
f.
5
7
b.
16.
c.
g. No real solution
h. ½
i.
5
7
j.
k.
l.
m.
n.
o.
–½
0.8
0.09
-0.5
0.04
No real solution
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
17
-20
8
19
No real solution
1/3
½
No real solution
-3/4
1/10
No real solution
-0.14
0.7
0.19
0.5
d.
e.
17.
19.
20.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
21.
18.
a.
b.
c.
d.
e.
22. 55
23. 26
24. 15
a.
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2.45
8.37
7.42
3.74
10.29
6.4
8.94
8.06
2.83
15.26
~ 17 ~
7
6
8
3
9
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25.
c. 4
d. 6
e. 7
29. 27
30. 37
31. 48
32.
a. Rational
b. Irrational
c. Irrational
d. Rational
e. Irrational
f. Rational
g. Irrational
h. Rational
i. Rational
j. Rational
33. c
34. a,b
35.
a. Rational
b. Irrational
c. Rational
d. Rational
e. Rational
f. Rational
g. Irrational
h. Irrational
i. Rational
j. Rational
36. a,c,d
37. a
38.
a. 7
b. 10
c. 3
d. 6
e. 8
f. 2
g. 2
h. 3
i. -3
j. -2
k. 5
l. 7
39.
mm2
40.
in3
a.
b.
c.
d.
e.
26.
27.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
8.83
2.65
7.94
5.39
6.48
11.75
17.32
12.17
4.58
7.21
28.
a. 4
b. 6
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41. a.
42.
a.
b.
c.
d.
e.
f.
g.
h.
x
b.
m2
i. -2
j. -4
k. 4
l. 9
43.
mm2
44.
in3
45. a.
x
9
7
2
3
8
3
4
4
b.
in2
Review answers
1. Yes
2. No
3. Yes
4. Yes
5. No
6. B
7. C
8. D
9. Irrational
10. rational
11. irrational
12. b
13. a
14. ~7
15. -2
16. 4
17. 5
18.
cm2
19.
=5.29
=5.48
=9.1
20.
21.
22. Expressions will vary
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