Math Grade 8 Unit 2
Roots and Exponents
Exercises
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Copyright © 2014 Pearson Education, Inc. 2
Grade 8 Unit 2: Roots and Exponents
CONTENTS
Exercises
LESSON 1:POWERS OF 2 ���������������������������������������������������������������������������� 4
LESSON 2:EXPONENTS ������������������������������������������������������������������������������ 5
LESSON 3:SQUARE ROOTS AND CUBE ROOTS ����������������������������� 10
LESSON 4:SIMPLIFYING EXPRESSIONS ����������������������������������������������� 14
LESSON 5:MULTIPLYING AND DIVIDING ������������������������������������������ 18
LESSON 6:PROPERTIES OF EXPONENTS ������������������������������������������� 22
LESSON 7:SCIENTIFIC NOTATION ������������������������������������������������������� 26
LESSON 8:Negative Exponents ������������������������������������������������������ 31
LESSON 9:PUTTING IT TOGETHER ����������������������������������������������������� 35
LESSON 13:CALCULATING WITH NOTATION ��������������������������������� 36
LESSON 14:RATIONAL NUMBERS ���������������������������������������������������������� 42
LESSON 15:ESTIMATING SQUARE ROOTS ������������������������������������������ 48
LESSON 16:Irrational Numbers ������������������������������������������������������ 52
LESSON 17:PUTTING IT TOGETHER ����������������������������������������������������� 58
Note: Some of these problems are designed to be delivered electronically.
Copyright © 2014 Pearson Education, Inc. 3
Grade 8 Unit 2: Roots and Exponents
LESSON 1: POWERS OF 2
Exercises
Exercises
•
Write your wonderings about roots and exponents.
•
Write a goal stating what you plan to accomplish in this unit.
•
Based on your previous work, write three things you will do differently during this
unit to increase your success.
Copyright © 2014 Pearson Education, Inc. 4
Grade 8 Unit 2: Roots and Exponents
LESSON 2: EXPONENTS
Exercises
EXERCISES
Evaluate the expressions in Exercises 1–7.
1. 34
 1
2.  
 2
4
3. 52
4. 43 • 62
2
 1
5.   • 33
 3
4
 1
6.   • 52
 4
7. 42 • 42 • 42 • 42
8. a.Exponents (especially powers of 2) are often used to express the memory capacity
of computers. For example, 512 megabytes (MB) of RAM could be expressed as 29.
Write the next four powers of 2 after 29 as whole numbers.
b. Describe another situation in which you have seen powers of 2 used.
Challenge Problem
9. Volume is often expressed as a cubic measurement, like cm3. How is this related to
raising a number to the power of 3? Why do you think this is called cubing a number?
Consider a cube with edge length 3 m. What is the volume of the cube?
What units are used for this volume?
Copyright © 2014 Pearson Education, Inc. 5
Grade 8 Unit 2: Roots and Exponents
LESSON 2: EXPONENTS
Exercises
ANSWERS
1. 81
3 • 3 • 3 • 3 = 81
1
16
2.
1 1 1 1
1
•
•
•
=
2 2 2 2
16
3. 25
5 • 5 = 25
4. 2,304
4 • 4 • 4 • 6 • 6 = 2,304
5. 3
1
1 1
• •3•3•3=
• 27 = 3
3 3
9
25
256
6.
1
25
1 1 1 1
•
•
•
•5•5=
• 25 =
4 4 4 4
256
256
7. 65,536
4 • 4 • 4 • 4 • 4 • 4 • 4 • 4 • = 65,536
Copyright © 2014 Pearson Education, Inc. 6
Grade 8 Unit 2: Roots and Exponents
LESSON 2: EXPONENTS
Exercises
answers
8.
Rubric
4 points
Response shows a sound approach, a correct answer, and good
communication of the process by which the answer was obtained.
3 points
Response shows a sound approach, but errors along the way may
result in an incorrect response. Work is clearly shown, but some of the
detail of the steps taken may be incomplete.
2 points
Response has an approach that is flawed but carried through
appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not
follow through well, with work shown being potentially sketchy. Or the
correct answer is shown, but no communication is given about how the
solution was obtained.
EXAMPLE OF A 4-POINT RESPONSE:
4 points
The next four powers of 2:
210 = 29 • 2 = 512 • 2 = 1,024
211 = 210 • 2 = 1,024 • 2 = 2,048
212 = 211 • 2 = 2,048 • 2 = 4,096
213 = 212 • 2 = 4,096 • 2 = 8,192
Answers will vary. Possible answer:
Powers of 2 can be used to talk about population growth.
For example, if a bacterium divides every hour, then after one hour,
the population is 2. After two hours, it is 4. After three hours, it is 8.
In general, the population after n hours is 2n.
Copyright © 2014 Pearson Education, Inc. 7
Grade 8 Unit 2: Roots and Exponents
LESSON 2: EXPONENTS
Exercises
answers
Challenge Problem
9.
Rubric
4 points
Response shows a sound approach, a correct answer, and good
communication of the process by which the answer was obtained.
3 points
Response shows a sound approach, but errors along the way may
result in an incorrect response. Work is clearly shown, but some of the
detail of the steps taken may be incomplete.
2 points
Response has an approach that is flawed but carried through
appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not
follow through well, with work shown being potentially sketchy. Or the
correct answer is shown, but no communication is given about how the
solution was obtained.
EXAMPLE OF A 4-POINT RESPONSE:
4 points
To find the volume of a cube, you multiply all the side lengths together.
For a cube with side length s, V = s • s • s. This is the same as raising s
to the power of 3. Raising a number to the power of 3 is called cubing
because it is how you calculate the volume of a cube. The volume of a
cube is s3. To find the units of the volume, the units of the length are
cubed as well.
So, a cube with edge length 3 m has a volume of 27 m3. To determine
this, first take the side length and cube it: 33 = 27. When you cube the
value, you also cube the units. So the volume unit must be m3.
Copyright © 2014 Pearson Education, Inc. 8
Grade 8 Unit 2: Roots and Exponents
LESSON 2: EXPONENTS
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
8
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
9
Copyright © 2014 Pearson Education, Inc. 9
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 3: SQUARE ROOTS AND CUBE ROOTS
Exercises
EXERCISES
1. Which number is not a perfect square?
A 4
C 48
B 25
D 64
2. If a square has an area of 144 square units, what is its side length?
3. If a square has an area of 0.25 square units, what is its side length?
For Exercises 4–7, solve each equation.
4. x2 = 25
5. x2 = 3,600
6. x2 = –16
7. x2 = 81
8. Give the volume and edge length of the large cube.
Challenge Problem
9. If you take the square root of a number and then take the cube root of the result,
you get 5. What is the number?
Copyright © 2014 Pearson Education, Inc. 10
Grade 8 Unit 2: Roots and Exponents
LESSON 3: SQUARE ROOTS AND CUBE ROOTS
Exercises
ANSWERS
1. C 48
Write the squares for the first few positive integers:
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
You can see from this list that 4, 25, and 64 are all perfect squares. For 48 to be a
perfect square, its square root must be an integer between 6 and 7. Such an integer
does not exist. So 48 cannot be a perfect square.
2. The side length is 12 units.
The formula for the area of a square is A = s2. Since 122 = 144, the side length for a
square with area 144 square units must be 12 units.
3. The side length is 0.5 unit.
2
The formula for the area of a square is A = s2. Since 0.5 = 0.25, the side length for a
square with area 0.25 square units must be 0.5 unit.
4. x = 5 or x = –5
Both 52 and (–5)2 equal 25, so x could be either of those values.
5. x = 60 or x = –60
Both 602 and (–60)2 equal 3,600, so x could be either of those values.
Copyright © 2014 Pearson Education, Inc. 11
Grade 8 Unit 2: Roots and Exponents
LESSON 3: SQUARE ROOTS AND CUBE ROOTS
Exercises
answers
6. No solution is possible.
Squaring a number always gives a positive remove forced carriage return the equation
given is not possible to solve.
7. x = 9 or x = –9
Both 92 and (–9)2 equal 81, so x could be either of those values.
8. Edge length = 5 units; Volume = 125 cubic units
The side length for the cube can be found by counting the number of cubes on one
side. The volume can be calculated using the volume formula: V = s3 = 53 = 125.
The side is given in units of the length of one side of the cube, which can just be called
a unit.Volume is the side length unit cubed—in this case, it is cubic units.
Challenge Problem
9. The number is 15,625.
To determine the number, you must take the inverse of each function. The inverse of
the cube root is the exponent 3, and the inverse of the square root is the exponent 2.
Applying these exponents to 5:
(53)2 = 5 • 5 • 5 • 5 • 5 • 5 =15,625
Copyright © 2014 Pearson Education, Inc. 12
Grade 8 Unit 2: Roots and Exponents
LESSON 3: SQUARE ROOTS AND CUBE ROOTS
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
8
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
9
Copyright © 2014 Pearson Education, Inc. 13
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 4: SIMPLIFYING EXPRESSIONS
EXERCISES
For Exercises 1–8, simplify each expression.
1. 6 • 6 • 6 • 6 • 6
2. 0.3 • 0.3 • 0.3 • 0.3
3. n • (4 • 4 • 4)
4. g + g + g + g + g + g + g
5
5. 13
132
6.
7.
3
343
(3 6)
3
 3
8.  
 7
2
Challenge Problem
9. Simply this expression.
 108 
4

 •6
3 
3
 4
7
  • 3
9
Copyright © 2014 Pearson Education, Inc. 14
Exercises
Grade 8 Unit 2: Roots and Exponents
LESSON 4: SIMPLIFYING EXPRESSIONS
Exercises
ANSWERS
1. 65 = 7,776
6 • 6 • 6 • 6 • 6 = 65 = 7,776
2. 0.34 = 0.0081
0.3 • 0.3 • 0.3 • 0.3 = 0.34 = 0.0081
3. n(43) = 64n
n • (4 • 4 • 4) = n(43) = 64n
4. 7g
g + g + g + g + g + g + g = 7g
5. 133 = 2,197
135 ÷ 132 =
13 • 13 • 13 • 13 • 13
= 13 • 13 • 13 = 133 = 2,197
13 • 13
6. 7
3
343 = 3 7 • 7 • 7 = 7
7. 6
The cube root is the inverse of the exponent 3, so the result is the original number.
8.
9
49
2
3 3 9
 3
  = • =
7
7 7 49
Copyright © 2014 Pearson Education, Inc. 15
Grade 8 Unit 2: Roots and Exponents
LESSON 4: SIMPLIFYING EXPRESSIONS
answers
Challenge Problem
9.
36 • 6 • 6 • 6 • 6
4 4 4
• • •3•3•3•3•3•3•3
9 9 9
6 • 1, 296
64
• 2,187
729
7, 776
= 40.5
192
Copyright © 2014 Pearson Education, Inc. 16
Exercises
Grade 8 Unit 2: Roots and Exponents
LESSON 4: SIMPLIFYING EXPRESSIONS
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
8
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
9
Copyright © 2014 Pearson Education, Inc. 17
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 5: MULTIPLYING AND DIVIDING
Exercises
EXERCISES
1. Which expression is equivalent to
A 33
39
?
33
C 312
B 36
D 3–6
For Exercises 2–8, write the expression as an integer (not a fraction) to a single power.
2.
63
67
3. 34 • 34 • 34
4.
11–7
11
5. 52 • 52 • 52 • 52
6.
7 –5
7 –7
7.
34 33
•
32 34
8.
8 4 83
•
8 2 85
9. Show that the following equation is true.
 23 • 7 8 
 25 • 7 –2 
–1
=
22
7 10
Challenge Problem
10. Simplify this expression.
46 • 4 8
53 • 58
Copyright © 2014 Pearson Education, Inc. 18
Grade 8 Unit 2: Roots and Exponents
LESSON 5: MULTIPLYING AND DIVIDING
ANSWERS
1. B 36
39 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3
=
= 3 • 3 • 3 • 3 • 3 • 3 = 36
3•3•3
33
2. 6–4
63
3–7
= 6−4
7 =6
6
3. 312
34 • 34 • 34 = 34+4+4 = 312
4. 11–8
11–7
= 11–7–1 = 11–8
11
5. 58
52 • 52 • 52 • 52 = 52+2+2+2 = 58
6. 72
7 –5
= 7(-5-(-7) ) =72
7 –7
7. 3
34 • 33
32 • 34
8.
33+ 4 37
=
= 37–6 = 31
32+ 4 36
=
8 4 +3 8 7
=
= 87–7 = 8 0
8 2+ 5 87
87
=1
87
8 4 • 83
8 2 • 85
Copyright © 2014 Pearson Education, Inc. =
19
Exercises
Grade 8 Unit 2: Roots and Exponents
LESSON 5: MULTIPLYING AND DIVIDING
Exercises
answers
9. Answers will vary. Possible answer:
 23 • 7 8 
 25 • 7 –2 
–1
 22   7 8  
=  5  •  −2  
 2   7  
−1
Challenge Problem
10.
46 • 4 8
53 • 58
43 • 4 4
53 • 54
47
57
Copyright © 2014 Pearson Education, Inc. 20
(
)(
)
=  23–5 • 7 8–( –2) 


−1
( )( )
=  2 –2 • 7 10 
−1
 7 10 
= 2 
2 
−1
=
22
7 10
Grade 8 Unit 2: Roots and Exponents
LESSON 5: MULTIPLYING AND DIVIDING
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
8
9
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
10
Copyright © 2014 Pearson Education, Inc. 21
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 6: PROPERTIES OF EXPONENTS
Exercises
EXERCISES
( )
7
1. If 4
3
= 4 x then what is the value of x?
A 4
C 21
B 10
D 28
2. If 137 • 139 = 13 y , then what is the value of y?
A 2
C 16
B 13
D 63
( )
2
4
3. Write four different expressions that are equivalent to 6 .
4. Write four different expressions that are equivalent to ( 9 • 2 ) .
4
5
10
5. Use the properties of exponents to show that 25 = 5 .
( ) = (2 )
4
6. Use the properties of exponents to show that 2
4
7. Explain why –2 is not equivalent to ( –2 ) • ( –2 ) .
2
Challenge Problem
8. Simplify this expression.
(x ) • (y )
(y ) • (x )
Copyright © 2014 Pearson Education, Inc. 4 5
3 7
4 5
7 3
22
2
3
6 2
.
Grade 8 Unit 2: Roots and Exponents
LESSON 6: PROPERTIES OF EXPONENTS
Exercises
ANSWERS
1. C 21
( 47 )
3
= 4x
47• 3 = 4 x
4 21 = 4 x
x = 21
2. C 16
137 • 139 = 13 y
137 + 9 = 13 y
1316 = 13 y
y = 16
3. Answers will vary. Ask a classmate to check your expressions.
( ) , (3
Possible answers: 68 , 64 • 64 , 62
4
2
• 22
)
4
4. Answers will vary. Ask a classmate to check your expressions.
Possible answers: 18 4 , 94 • 24 , 38 • 4 2
( )
5. 255 = 52
6.
Copyright © 2014 Pearson Education, Inc. (2 )
4 3
5
= 510
( )
= 212 = 26•2 = 26
2
23
Grade 8 Unit 2: Roots and Exponents
LESSON 6: PROPERTIES OF EXPONENTS
Exercises
answers
7.
Rubric
4 points
Response shows a sound approach, a correct answer, and good
communication of the process by which the answer was obtained.
3 points
Response shows a sound approach, but errors along the way may
result in an incorrect response. Work is clearly shown, but some of the
detail of the steps taken may be incomplete.
2 points
Response has an approach that is flawed but carried through
appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not
follow through well, with work shown being potentially sketchy. Or the
correct answer is shown, but no communication is given about how the
solution was obtained.
EXAMPLE OF A 4-POINT RESPONSE:
4 points
The expression –24 means the negative of 24, which is –16.
The expression (–2)2 is the square of –2, which is 4.
So (–2)2 • (–2)2 = 4 • 4 = 16.
Challenge Problem
8.
(x ) • (y )
(y ) • (x )
4 5
3 7
4 5
7 3
x 20 • y 21
y
Copyright © 2014 Pearson Education, Inc. 20
•x
21
=
y
x
24
Grade 8 Unit 2: Roots and Exponents
LESSON 6: PROPERTIES OF EXPONENTS
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
8
Copyright © 2014 Pearson Education, Inc. 25
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 7: SCIENTIFIC NOTATION
Exercises
EXERCISES
1. Which number is in scientific notation?
A 27.2 • 108
B 1.4 • 56
C 0.907 • 10–12
D 5.001 • 10–25
2. Which is the scientific notation form of 409.9 × 109?
A 4.099 • 107
B 4.099 • 1011
C 0.000004099 D 0.0000004099
For Exercises 3–5, write the measurement in scientific notation.
3. Half-life of a helium-7 atom: 0.00000000000000000000304 sec
4. Water molecules in 1 cup of water: 7,500,000,000,000,000,000,000,000 molecules
5. Circumference of Neptune’s orbit: 17,562,000,000 mi.
For Exercises 6–7, write the number in scientific notation.
6. 0.00444 • 1015
7. 0.75 • 10–42
For Exercises 8–9, write the number in standard notation.
8. 8.068 • 10–8
9. 1.00001 • 1013
For Exercises 10–12, compare the two expressions. Choose <, >, or =.
10. 4.5 • 1020
11. 72,000,000,000
12. 3.2 • 10–8
Copyright © 2014 Pearson Education, Inc. 4.5 • 10–20
2.5 • 1011
1.2 • 10–7
26
Grade 8 Unit 2: Roots and Exponents
LESSON 7: SCIENTIFIC NOTATION
Exercises
exercises
Challenge Problem
13. The micrometer, nanometer, and picometer are units used to measure very small objects.
Unit
Length
micrometer (µm)
10–6 m
(1 millionth of a meter)
nanometer (nm)
10–9 m
(1 billionth of a meter)
picometer (pm)
10–12 m
(1 trillionth of a meter)
a. The length of an amoeba proteus is 500 µm. What is this length in meters?
Give your answer in scientific notation.
b. The diameter of a glucose molecule is 900 pm. What is the diameter of a
glucose molecule in micrometers?
c. A measles virus has a diameter of 220 nm. What is the diameter of a measles
virus in meters?
Copyright © 2014 Pearson Education, Inc. 27
Grade 8 Unit 2: Roots and Exponents
LESSON 7: SCIENTIFIC NOTATION
Exercises
ANSWERS
1. D 5.001 • 10–25
A number written in scientific notation has two factors: one that is greater than or
equal to 1 and less than 10, and one that is a power of 10. Answer D is the only one
with both factors written in this way.
2. B 4.099 • 1011
409.9 • 109 = (4.099 • 102) • 109 = 4.099 • (102 • 109) = 4.099 • 1011
The first factor is between 1 and 10, and the second factor is a power of 10, so it is
written properly in scientific notation.
3. 3.04 • 10–21 sec
4. 7.5 • 1024 molecules
5. 1.7562 • 1010 mi
6. 4.44 • 1012
7. 7.5 • 10–43
8. 0.00000008068
9. 10,000,100,000,000
10. 4.5 • 1020
>
11. 72,000,000,000
12. 3.2 • 10–8
Copyright © 2014 Pearson Education, Inc. <
4.5 • 10–20
<
2.5 • 1011
1.2 • 10–7
28
Grade 8 Unit 2: Roots and Exponents
LESSON 7: SCIENTIFIC NOTATION
Exercises
answers
Challenge Problem
13. Rubric
4 points
Response shows a sound approach, a correct answer, and good
communication of the process by which the answer was obtained.
3 points
Response shows a sound approach, but errors along the way may
result in an incorrect response. Work is clearly shown, but some of the
detail of the steps taken may be incomplete.
2 points
Response has an approach that is flawed but carried through
appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not
follow through well, with work shown being potentially sketchy. Or the
correct answer is shown, but no communication is given about how the
solution was obtained.
EXAMPLE OF A 4-POINT RESPONSE:
4 points
a. The length of an amoeba proteus is 5.00 • 10–4 m.
1 µm = 10–6 m, which can be rewritten as 106 µm = 1 m and used to
convert the length. Then rewrite in scientific notation:
500 µm = 500 • 10–6 m = (5.00 • 102) • 10–6 m = 5.00 • 10–4 m
b. The diameter of a glucose molecule is 9.00 • 10–4 µm.
1 µm = 10–6 m can be rewritten as 106 µm = 1 m, and 1 pm = 10–12 m
can be rewritten as 1012 pm = 1 m. Setting these equal
to each other gives 1012 pm = 106 µm, which can be rewritten as
(10–6)1012 pm = (10–6)106 µm  106 pm = 1 µm and used
to convert the length. Then rewrite in scientific notation:
900 pm = 900 • 10–6 µm = (9.00 • 102) • 10–6 µm = 9.00 • 10–4 µm
c. The diameter of a measles virus is 2.20 • 10–7 m.
1 nm = 10–9 m, which can be rewritten as 109 nm = 1 m and used to
convert the length. Then rewrite in scientific notation:
220 nm = 220 • 10–9 m = (2.20 • 102) • 10–9 m = 2.20 • 10–7 m
Copyright © 2014 Pearson Education, Inc. 29
Grade 8 Unit 2: Roots and Exponents
LESSON 7: SCIENTIFIC NOTATION
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
8
9
10
11
12
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
13 a.
13 b.
13 c.
Copyright © 2014 Pearson Education, Inc. 30
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 8: Negative Exponents
Exercises
EXERCISES
For Exercises 1–8, find the value of the expression.
1. 7–2
2. 10
3. 10–7
4. 3–3
5. 4–8 • 48
6. 8–1
7.
(2 )
8.
(3 )
–3 –2
–2 –2
9. Circle the expressions that are equivalent to 4–7 • 43.
4–4
4–21
410
1
44
43
47
8–4
(–25)4
5–8
10. Circle the expressions that are equivalent to (5–2)4.
4
 1
 2 
5
254
52
5–4 • 5–4
Challenge Problem
11. Give an equation of the form an = b that satisfies the given criteria.
If it is not possible, explain why.
a.
b.
c.
d.
Copyright © 2014 Pearson Education, Inc. a is negative, n is negative, and b is positive.
a is positive, n is positive, and b is positive.
a is negative, n is positive, and b is negative.
a is positive, n is negative, and b is negative.
31
Grade 8 Unit 2: Roots and Exponents
LESSON 8: Negative Exponents
Exercises
ANSWERS
1
49
1.
7 –2 = 49 –1 =
1
49
2. 1
10 = 1
1
10, 000, 000
3.
10 –7 = 10, 000, 000 –1 =
1
10, 000, 000
1
27
4.
3–3 = 27 –1 =
1
27
5. 1
4–8 • 48 = 4–8 + 8 = 40 = 1
6.
1
8
8 –1 =
1
8
7. 64
(2 )
–3 –2
Copyright © 2014 Pearson Education, Inc. = 26 = 64
32
Grade 8 Unit 2: Roots and Exponents
LESSON 8: Negative Exponents
Exercises
answers
8. 81
(3 )
–2 –2
= 34 = 81
9. 4–4 1
43
47
44
4
 1
10.  2  5–4 • 5–4 5–8
5 
Challenge Problem
11. a.Answers will vary. All answers in which a < 0, n < 0 and even, and b > 0 are correct.
−2
Possible answer: ( –2 ) =
1
4
b.Answers will vary. All answers in which a > 0, n > 0, and b > 0 are correct.
2
Possible answer: 2 = 4
c.Answers will vary. All answers in which a < 0, n > 0 and odd, and b < 0 are correct.
Possible answer: ( –2 ) = –8
3
d.This situation is not possible because a positive number raised to any power
is positive.
Copyright © 2014 Pearson Education, Inc. 33
Grade 8 Unit 2: Roots and Exponents
LESSON 8: Negative Exponents
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
8
9
10
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
11 a.
11 b.
11 c.
11 d.
Copyright © 2014 Pearson Education, Inc. 34
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 9: PUTTING IT TOGETHER
Exercises
•
Read through your work on the Self Check task and think about your other work
in this lesson.
•
Write what you have learned.
•
What would you do differently if you were starting the Self Check task now?
•
Record your ideas. Keep track of any strategies you have learned.
•
Complete any exercises that you have not finished from this unit.
Copyright © 2014 Pearson Education, Inc. 35
Grade 8 Unit 2: Roots and Exponents
LESSON 13: CALCULATING WITH NOTATION
Exercises
Exercises
1. In 2010 in the United Kingdom, 129,000,000,000 text messages were sent.
That same year in the Netherlands, 11,000,000,000 text messages were sent.
a. Estimate each number as a single digit times a power of 10.
b. Use your estimates to help you write a statement comparing the number of
text messages sent in the two countries.
2. In 2010, the United States population was 309,975,000, and the population
of Australia was 22,421,417.
a. Estimate each number as a single digit times a power of 10.
b. Use your estimates to help you write a statement comparing the populations
of the two countries.
3. The diameter of a fluorine ion is 0.000000000038 m. The diameter of a small grain of
sand is about 0.00002 m.
a. Estimate each number as a single digit times a power of 10.
b. Use your estimates to help you write a statement comparing the
two diameters.
4. The table shows the land area of six countries in square kilometers.
Country
Russia
United States
Area (km2)
17,075,200
9,826,630
Kenya
582,650
Uruguay
176,220
Haiti
27,750
Singapore
693
Monaco
2
a. Estimate each number as a single digit times a power of 10.
b. Write four statements comparing the areas of the countries in the table.
Copyright © 2014 Pearson Education, Inc. 36
Grade 8 Unit 2: Roots and Exponents
LESSON 13: CALCULATING WITH NOTATION
Exercises
exercises
Challenge Problem
5. About 111,041,000 people in the United States tuned in to watch Super Bowl XLV in
2011. The 2011 Academy Awards were watched by about one-third as many people.
a. Rewrite the number of people who watched the Super Bowl
in scientific notation.
b. Round the first factor to the hundredths place. Then use that number to
estimate how many people watched the Academy Awards.
Copyright © 2014 Pearson Education, Inc. 37
Grade 8 Unit 2: Roots and Exponents
LESSON 13: CALCULATING WITH NOTATION
Exercises
ANSWERS
1. a. In the United Kingdom, about 1 • 1011 text messages were sent, and in the
Netherlands, about 1 • 1010 text messages were sent.
b. Answers will vary. Two possible answers:
•
The number of text messages sent in the United Kingdom was about
10 times the number sent in the Netherlands.
1• 1011
= 10
1• 1010
•
1
The number of text messages sent in the Netherlands was about
the
10
number sent in the United Kingdom.
1
1• 1010
11 =
10
1• 10
2. a. T
he U.S. population was about 3 • 108, and the Australian population was
about 2 • 107.
b. Answers will vary. Two possible answers:
•
The U.S. population was about 15 times the Australian population.
3 • 10 8 3
= • 10 = 15
2 • 107 2
•
The Australian population was about
1
the U.S. population.
15
2 • 107 2 1
1
•
=
8 =
3 10 15
3 • 10
3. a. A
fluorine ion is about 4 • 10–11 m in diameter. A grain of sand is about
2 • 10–5 m in diameter.
b. Answers will vary. Two possible answers:
•
The diameter of a grain of sand is about 500,000 times the diameter
of a fluorine ion.
2 • 10 –5 1
• 106 = 0.5 • 1, 000, 000 = 500, 000
–11 =
2
4 • 10
Copyright © 2014 Pearson Education, Inc. 38
Grade 8 Unit 2: Roots and Exponents
LESSON 13: CALCULATING WITH NOTATION
Exercises
answers
•
1
The diameter of a fluorine ion is about
the diameter
500,
000
of a grain of sand.
4 • 10 –11
2
1
–6
=
=
–5 = 2 • 10
1,
000,
000
500,
000
2 • 10
4.
Copyright © 2014 Pearson Education, Inc. Rubric
4 points
Response shows a sound approach, a correct answer, and good
communication of the process by which the answer was obtained.
3 points
Response shows a sound approach, but errors along the way may
result in an incorrect response. Work is clearly shown, but some of the
detail of the steps taken may be incomplete.
2 points
Response has an approach that is flawed but carried through
appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not
follow through well, with work shown being potentially sketchy. Or the
correct answer is shown, but no communication is given about how the
solution was obtained.
39
Grade 8 Unit 2: Roots and Exponents
LESSON 13: CALCULATING WITH NOTATION
Exercises
answers
EXAMPLE OF A 4-POINT RESPONSE:
4 points
a.
Country
Area (km2)
Russia
2 × 107
United States
1 × 107
Kenya
6 × 105
Uruguay
2 × 105
Haiti
3 × 104
Singapore
7 × 102
Monaco
2 × 100
b. Answers will vary. Possible answers:
•
The area of Russia is about twice the area of the
United States.
•
The area of the United States is about 50 times the area
of Uruguay.
1
The area of Haiti is about
the area of Kenya.
20
The area of Singapore is about 350 times the area
of Monaco.
•
•
Challenge Problem
5. a. 111,041,000 = 1.11041 • 108
b. About 37,000,000 people watched the Academy Awards.
Rounding 1.11041 • 108 to the hundredths place gives 1.11 • 108
Multiply this number by one-third to estimate the number
of Academy Award watchers:
Copyright © 2014 Pearson Education, Inc. 1
1.11
• 1.11• 10 8 =
• 10 8 = 37, 000, 000
3
3
40
Grade 8 Unit 2: Roots and Exponents
LESSON 13: CALCULATING WITH NOTATION
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1 a.
1 b.
2 a.
2 b.
3 a.
3 b.
4 a.
4 b.
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
5 a.
5 b.
Copyright © 2014 Pearson Education, Inc. 41
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 14: RATIONAL NUMBERS
Exercises
EXERCISES
1. Write
1
in decimal form.
3
2. Which fraction will have a decimal form that is a repeating decimal?
A
1
5
B
3
8
C
7
12
D
19
20
3. a. Use your calculator to write each fraction as a decimal.
1
11
2
11
3
11
4
11
5
11
=
=
=
=
=
b. Describe the pattern in the results from part a.
c. Predict the decimal forms of
4. Explain why the decimal form of
Copyright © 2014 Pearson Education, Inc. 42
7
10
and
. Then check your prediction.
11
11
1
7
repeats, but the decimal form of
terminates.
14
14
Grade 8 Unit 2: Roots and Exponents
LESSON 14: RATIONAL NUMBERS
Exercises
exercises
For Exercises 5–7, write each decimal as a fraction.
5. 0.099
6. 0.27
7. 0.53
Challenge Problem
8. Did you know that the repeating decimal 0.9 is equal to 1? You can show that this is
true in two different ways.
a. Use the fact that 0.3 =
1
to help you show that 0.9 = 1 .
3
b. Use the method you learned for changing a repeating decimal to a fraction to
show that 0.9 = 1 .
Copyright © 2014 Pearson Education, Inc. 43
Grade 8 Unit 2: Roots and Exponents
LESSON 14: RATIONAL NUMBERS
Exercises
ANSWERS
1. 0.3 =
2. C 1
3
7
12
You can use a calculator to determine each fraction in decimal form:
1
= 0.2
5
3
= 0.375
8
7
= 0.583
12
19
= 0.95
20
3. a.
1
= 0.09
11
2
= 0.18
11
3
= 0.27
11
4
= 0.36
11
5
= 0.45
11
b.The repeating digits of each decimal are equal to 9 times the numerator
of the fraction.
7
c. = 0.63
11
10
= 0.90
11
Copyright © 2014 Pearson Education, Inc. 7
is 63.
11
10
10 • 9 = 90, so the repeating decimal in
is 90.
11
7 • 9 = 63, so the repeating decimal in
44
Grade 8 Unit 2: Roots and Exponents
LESSON 14: RATIONAL NUMBERS
Exercises
answers
1
in decimal form is 0.0714285 . Looking at the
14
long division algorithm can help you understand why it is a repeating decimal:
4. Answers will vary. Possible answer:
0. 0 7
14 ) 1. 0 0
98
2
1
142857
000000
0
4
60
56
40
28
120
112
80
70
100
98
2
The dividends created by the remainders for the first few steps are 20, 60, 40, 120,
80, and 100. None of these numbers are multiples of 14, so they all result in a
remainder. The last remainder shown in this worked-out example is 2, which creates
a dividend of 20. The dividends will repeat themselves again and will continue in this
manner infinitely.
1
Since 14 is a multiple of 7, this fraction can be reduced to , which is a terminating
2
decimal.
Copyright © 2014 Pearson Education, Inc. 45
Grade 8 Unit 2: Roots and Exponents
LESSON 14: RATIONAL NUMBERS
Exercises
answers
5.
99
1, 000
6.
3
11
7.
8
15
Challenge Problem
1
8. a. 1 = 3 • = 3 • 0.3 = 0.9
3
10 x = 9.9
b.
–
x = 0.9
9x = 9
x=1
Copyright © 2014 Pearson Education, Inc. 46
Grade 8 Unit 2: Roots and Exponents
LESSON 14: RATIONAL NUMBERS
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3 a.
3 b.
3 c.
4
5
6
7
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
8 a.
8 b.
Copyright © 2014 Pearson Education, Inc. 47
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 15: ESTIMATING SQUARE ROOTS
Exercises
EXERCISES
For Exercises 1–7, estimate the value of each square root. Then explain your method for
estimating each. What two whole numbers is each square root value between?
1.
3
2.
7
3.
10
4.
6
5.
11
6.
48
7.
63
Challenge Problem
8. Estimate the value of 142 . Explain your method. What two whole numbers is the
square root value between?
Copyright © 2014 Pearson Education, Inc. 48
Grade 8 Unit 2: Roots and Exponents
LESSON 15: ESTIMATING SQUARE ROOTS
Exercises
ANSWERS
3 ≈ 1.732
1.
1 < 3 < 4 , which means 1 < 3 < 2 .
1.72 = 2.89 and 1.82 = 3.24, which means 1.7 < 3 < 1.8 .
1.732 = 2.9929 and 1.742 = 3.0276, which means 1.73 < 3 < 1.74 .
1.7322 = 2.99824 and 1.7332 = 3.003289, which means 1.732 < 3 < 1.733 .
Since 1.7322 is closer to 3, 1.732 is a better estimate for
3.
The square root value is between 1 and 2, but closer to 2.
7 ≈ 2.645
2.
Solution method is similar to that for Exercise 1.
The square root value is between 2 and 3, but closer to 3
10 ≈ 3.162
3.
Solution method is similar to that for Exercise 1.
The square root value is between 3 and 4, but closer to 3
6 ≈ 2.449
4.
Solution method is similar to that for Exercise 1.
The square root value is between 2 and 3, but slightly closer to 2
11 ≈ 3.316
5.
Solution method is similar to that for Exercise 1.
The square root value is between 3 and 4, but closer to 3
Copyright © 2014 Pearson Education, Inc. 49
Grade 8 Unit 2: Roots and Exponents
LESSON 15: ESTIMATING SQUARE ROOTS
Exercises
answers
48 ≈ 6.928
6.
Since
49 is 7, you can estimate that
48 is almost 7.
The square root value is between 6 and 7, but much closer to 7.
63 ≈ 7.937
7.
Since
64 is 8, you can estimate that
63 is almost 8.
The square root value is between 7 and 8, but much closer to 8.
Challenge Problem
142 = 11.916
8.
Since
144 is 12, you can estimate that
142 is almost 12.
The square root value is between 11 and 12, but much closer to 12.
Copyright © 2014 Pearson Education, Inc. 50
Grade 8 Unit 2: Roots and Exponents
LESSON 15: ESTIMATING SQUARE ROOTS
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
8
Copyright © 2014 Pearson Education, Inc. 51
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 16: Irrational Numbers
Exercises
EXERCISES
1. Which number is irrational?
A
81 B
1
3
C 0.45 D
13
D −
17
47
2. Which number is irrational?
A 9π B 0.4321
C
3. Which two consecutive whole numbers is
3
27 70 between?
For Exercises 4–7, compare each pair of numbers and choose <, >, or =.
4.
8
5. 1.1
2.9
π
3
6. 2π
37
7. – π 2
−
Copyright © 2014 Pearson Education, Inc. 59
6
52
Grade 8 Unit 2: Roots and Exponents
LESSON 16: Irrational Numbers
Exercises
exercises
8. Place
0
5
33 5
,
, and
1
10 at 10
their approximate locations on the number line.
2
3
1
π
3
2× 3
4
5
1
π
3
1
Challenge Problem
5
π
2× 3
1
3
35
2
π
2
35
1
9. Place 3 π , 2 × 3 , and 23 at their approximate locations on the number line.
3
3
5
2
2× 3
3
5
2
3
0
1
2
3
4
5
2
Copyright © 2014 Pearson Education, Inc. 53
Grade 8 Unit 2: Roots and Exponents
LESSON 16: Irrational Numbers
Exercises
ANSWERS
1. D 13
All the numbers except for 13 can be written as fractions or whole numbers,
so they must be rational numbers.
81 = 9
3
is already a fraction.
8
0.45 =
5
11
2. A 9π
All the numbers except for 9π can be written as fractions or whole numbers,
so they must be rational numbers.
0.4321=
4,321
10, 000
3
27 = 3
−
17
is already a fraction.
47
70 is between 8 and 9.
3.
82 = 64 and 92 = 81
So, 8 <
4.
8
< 2.9
5. 1.1
>
6. 2π
>
Copyright © 2014 Pearson Education, Inc. 70 < 9.
π
3
37
54
Grade 8 Unit 2: Roots and Exponents
LESSON 16: Irrational Numbers
Exercises
answers
7. – π 2
8.
<
−
59
6
Rubric
4 points
Response shows a sound approach, a correct answer, and good
communication of the process by which the answer was obtained.
3 points
Response shows a sound approach, but errors along the way may
result in an incorrect response. Work is clearly shown, but some of the
detail of the steps taken may be incomplete.
2 points
Response has an approach that is flawed but carried through
appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not
follow through well, with work shown being potentially sketchy. Or the
correct answer is shown, but no communication is given about how the
solution was obtained.
EXAMPLE OF A 4-POINT RESPONSE:
4 points
The number line is drawn with tenth markings. To place the numbers
accurately, you need to estimate them to tenths.
1 <
4 , which means 1 <
3 <
3 < 2.
3 < 1.8.
1.72 = 2.89 and 1.82 = 3.24, which means 1.7 <
3 should be placed between 1.7 and 1.8.
Using a similar method, you find that 2.2 <
3.1 < 10 < 3.2.
3
0
Copyright © 2014 Pearson Education, Inc. 5
1
55
2
5 < 2.3 and
10
3
4
5
Grade 8 Unit 2: Roots and Exponents
LESSON 16: Irrational Numbers
Exercises
Challenge Problem
9.
1
π ≈ 1.047
3
2 × 3 ≈ 3.464
5
= 1.6
3
2 ≈ 1.414
1
π
3
0
Copyright © 2014 Pearson Education, Inc. 1
5
3
2
2× 3
2
3
56
4
5
Grade 8 Unit 2: Roots and Exponents
LESSON 16: Irrational Numbers
Exercises
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess
your work. For each exercise, place a check mark in the column that best describes how
you did on that exercise.
Exercise
Number
I was confused, but
now I get it.
Yes! I got it.
I need help!
1
2
3
4
5
6
7
8
Challenge Problem
Exercise
I gave it a try, but I’m not sure
I did it right.
9
Copyright © 2014 Pearson Education, Inc. 57
I did it, and my answer
makes sense.
Grade 8 Unit 2: Roots and Exponents
LESSON 17: PUTTING IT TOGETHER
Exercises
•
Read through your work on the Self Check task and think about your other work
in this lesson.
•
Write what you have learned.
•
What would you do differently if you were starting the Self Check task now?
•
Record your ideas. Keep track of any strategies you have learned.
•
Complete any exercises that you have not finished from this unit.
Copyright © 2014 Pearson Education, Inc. 58