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65©
Cu
rriculu
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ssociates, LL
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Copyin
g is not perm
itted.
Practice Lesson 15 Num
erical Expressions with Exponents
Unit 3
Practice and Prob
lem Solvin
gU
nit 3 Expressions an
d Equations
Key
B Basic M Medium C Challenge
©Curriculum Associates, LLC Copying is not permitted. 161Lesson 15 Numerical Expressions with Exponents
Lesson 15
Name: Numerical Expressions with Exponents
Vocabularypower of ten a number
that can be written as a
product of tens
100 and 102 are powers
of ten
exponent the number
in a power that shows
how many times to
multiply the base by
itself
In the expression 102, the
exponent is 2 and the
base is 10
Prerequisite: Multiplying by a Power of 10
Study the example showing how to multiply by a power of 10. Then solve problems 1–7.
1 By how many factors of 10 did you multiply 0 008? Why?
2 Consider the expression 103 3 0 006
a. What is the exponent in the power of 10?
b. How many factors of 10 are in 103?
c. How do your answers to the last two questions relate to one another?
d. What is the value of 103 3 0 006?
Example
Find 102 3 0 008
Break 102 into a product of 10s and multiply
102 3 0 008 5 100 3 0 008 5 10 3 10 3 0.008
5 10 3 0.08
5 0 8
This means that 102 3 0 008 5 0 8
161161
I multiplied 0.008 by two factors of 10 because
102 5 100 5 10 3 10.
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3
6
Possible answer: They are the same because
the exponent tells you how many times to
multiply 10 by itself.
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©Curriculum Associates, LLC Copying is not permitted.162 Lesson 15 Numerical Expressions with Exponents
Solve.
3 Complete the equations showing powers of 10 using exponents
a. 3 3 1,000 5 3 3 5
b. 0 07 3 100 5 0 07 3 5
c. 0 009 3 5 0 009 3 102 5
4 Find each product Explain how the place value of the digit 6 changes as the exponent changes
a. What is 0 006 3 101?
b. What is 0 006 3 102?
c. What is 0 006 3 103?
5 Describe the similarities and diff erences between 0 008 3 100 and 0 008 3 102
6 What power of 10 can you multipy 0 02 by to get a product of 20? Explain your answer
7 What is the product 4 3 1,000? Explain how you know
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103
102
3,000
0.9
7
100
0.06
0.6
6
Possible answer: Similarities: Both expressions have a value of 0.8; both expressions
represent 0.008 multiplied by the same power of 10. Difference: The power of 10 is in
standard form in the first expression and is shown with an exponent in the second.
1,000 or 103; Possible explanation: I can multiply 0.02 by 10 to get 0.2, then by 10 again
to get 2, and by one more factor of 10 to get 20. So that is a total of three factors of 10,
or 1,000.
4,000; Possible explanation: There are three factors of 10 in 1000, so you multiply 4 by
10 three times to get 4,000.
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Possible explanation: As the exponent increases by 1, the place value of 6 is
10 times as great.
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©C
urricu
lum
Associates, L
LC
C
opying is n
ot permitted.
Practice and Prob
lem Solvin
gU
nit 3 Expressions an
d Equations
Unit 3
Practice Lesson 15 Num
erical Expressions with Exponents
©Curriculum Associates, LLC Copying is not permitted. 163Lesson 15 Numerical Expressions with Exponents
Name: Lesson 15
Write and Evaluate Expressions with Exponents
Study the example problem showing how to write and evaluate expressions with exponents. Then solve problems 1–9.
1 What does it mean to say that the amount of money from the previous month is quadrupled?
2 Represent the problem with repeated multiplication
Month Amount Saved (in dollars)
1 4
2 4 • 5 16
3
4
3 Write an expression using an exponent to represent the amount of money Adrian will have saved by month 4
4 What is the value of the expression you wrote in problem 3?
Example
Adrian wants to buy a skateboard that costs $85 After 1 month, he has $4 in savings and plans to quadruple the amount he has saved each month for 4 months Will Adrian have enough money to buy the skateboard in 4 months?
Month 1 Month 2 Month 3 Month 4
4 4 • 4 5 16 16 • 4 5 64 64 • 4 5 256
Adrian will have enough money to buy the skateboard in 4 months He will have $171 more than he needs
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When you quadruple an amount, you multiply by 4.
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256
4
4 • 4 • 4 5 64
4 • 4 • 4 • 4 5 256
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©Curriculum Associates, LLC Copying is not permitted.164 Lesson 15 Numerical Expressions with Exponents
Solve.
Use the following situation to solve problems 5–7.
Five students received the same text message at 9:00 AM. Each of them sent the message to 5 more students at 10:00 AM. Each of those students sent the message to 5 more students at 11:00 AM.
5 Represent the situation with exponential expressions Simplify the expressions
Time That Message Is Received
Number of Students ReceivingText Message
9:00 AM 51 5 5
10:00 AM
11:00 AM
6 If the pattern continues, how many students will receive the text message at noon? Explain how to use the pattern to fi nd the answer
7 If the pattern continues, at what time will 15,625 students receive the text message? Explain how you know
8 Write and simplify an expression to represent 63
9 Chin says that the value of 25 is 10 Explain what Chin did wrong and fi nd the correct value
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52 5 25
53 5 125
625 students; Possible explanation: The number of students who receive the text
message is multiplied by 5 each hour. So the number of students who will receive
the text message at noon is 54, which equals 625.
2:00 PM; Possible explanation: 54, or 625, students receive the message at noon. So at
1:00 PM, 55, or 3,125, students will receive the message. That means that at 2:00 PM, 56,
or 15,625, students will receive the message.
6 • 6 • 6 5 216
Chin multiplied 2 • 5 rather than multiplying 2 • 2 • 2 • 2 • 2; the value is 32.
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67©
Cu
rriculu
m A
ssociates, LL
C
Copyin
g is not perm
itted.Practice an
d Problem
Solving
Unit 3 Exp
ressions and Eq
uations Unit 3
Practice Lesson 15 Num
erical Expressions with Exponents
©Curriculum Associates, LLC Copying is not permitted.166 Lesson 15 Numerical Expressions with Exponents
Solve.
5 What is the value of 4 1 23 • 3?
Show your work.
Solution:
6 What is the value of 42 ·· 2 ? Describe the steps you took
to fi nd your answer
7 Darren and Barb each tried to evaluate 62 1 4 4 2
Darren Barb
62 1 4 4 2 62 1 4 4 2 5 36 1 4 4 2 5 36 1 4 4 2 5 40 4 2 5 36 1 2 5 20 5 38
Who evaluated the expression correctly? Explain what the other student did wrong
8 Use the numbers 8, 6, and 2 and one operation to write an expression that includes an exponent and has a value of 8 Use each number only once
9 Show where to place parentheses in the expression 4 1 32 • 5 2 2 so that the value of the expression is 31
4 1 32 • 5 2 2
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8; Find the value of 42 first because the order of operations tells you to evaluate
exponential expressions before you divide. Then divide 16 by 2.
Barb; Darren added 36 1 4 in the second step, but the order of operations tells
you to divide before adding. So he should have divided 4 by 2 before adding 36.
The value of 4 1 23 • 3 is 28.
4 1 23 • 3 5 4 1 8 • 35 4 1 245 28
26 4 8
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©Curriculum Associates, LLC Copying is not permitted. 165Lesson 15 Numerical Expressions with Exponents
Name: Lesson 15
Evaluate Expressions with Exponents
Study the example problem showing how to evaluate expressions with exponents. Then solve problems 1–9.
1 Explain why you must simplify 32 fi rst
2 Diallo says that the value of 12 2 32 is 81 How did he get that answer?
3 Maggie says that if the expression was 12 4 32, you would divide before simplifying 32 Is she right? Explain
4 Suppose the expression was (12 2 3)2 Would you still simplify 32 fi rst? Explain
Example
Follow the order of operations to simplify 12 2 32
First find 32 32 5 3 • 3 5 9
Then subtract 9 from 12 12 2 9 5 3
This means that:
12 2 32 5 12 2 9 5 3
The value of the expression is 3
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Possible answer: The order of operations says that you must simplify exponential
expressions before subtracting.
Possible answer: He did not follow the order of operations. He worked from left to
right. He subtracted 3 from 12 to get 9 and then found 92.
No; Using the order of operations, you simplify exponential expressions before
adding, subtracting, multiplying, or dividing.
No; Using the order of operations, you simplify expressions inside parentheses first.
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68©
Cu
rriculu
m A
ssociates, LL
C
Copyin
g is not perm
itted.Practice an
d Problem
Solving
Unit 3 Exp
ressions and Eq
uations
Unit 3
Practice Lesson 15 Num
erical Expressions with Exponents
©Curriculum Associates, LLC Copying is not permitted.168 Lesson 15 Numerical Expressions with Exponents
6 Students are getting signatures for a petition to increase sports activities at the community center The number of signatures they get each day is 3 times as many as the day before The expression 36 represents the number of signatures they got on the sixth day How many signatures did they get on the fi rst day?
A 3
B 6
C 18
D 729
Betsy chose B as the correct answer How did she get that answer?
Solve.
4 Which expression shows the fi rst step in evaluating
2 1 7 • 12 ·· 6 2 32?
A 2 1 84 ·· 6 2 32
B 9 • 12 ·· 6 2 32
C 2 1 7 • 12 ·· 6 2 9
D 2 1 7 • 2 2 32
5 Students at a cooking school made a supersized rectangular pizza for a class party Lupita cut the pizza into 3 equal pieces Then she cut each piece into 3 equal parts two more times Lupita needs 27 pieces of pizza Does she have enough pieces yet? Explain how you know
What do the base and the exponent represent?
What operation is done first in the order of operations?
How can you use exponents to help you solve this problem?
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Yes; 33 5 3 3 3 3 3 5 27. So Lupita has 27 pieces.
She confused the base with the exponent in 36.
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©Curriculum Associates, LLC Copying is not permitted. 167Lesson 15 Numerical Expressions with Exponents
Name: Lesson 15
Numerical Expressions with Exponents
Solve the problems.
3 Beth is making a beanbag seat in the shape of a cube Each side of the seat is 2 feet long Beth needs to fi nd the volume of the seat so that she can buy the correct amount of beans Beans are sold in bags that hold 2 cubic feet of beans How many bags of beans should Beth buy?
Show your work.
Solution:
2 Look at the expression
4 • (12 2 8) 1 23
Tell whether each statement about the expression is True or False.
a. The last step in evaluating the expression is to simplify 23 u True u False
b. The value of 23 is 6 u True u False
c. The first step in evaluating the expression is to subtract 12 2 8 u True u False
d. The value of the expression is 48 u True u False
1 What is the value of 0 9 • 102?
A 0 09
B 0 9
C 9
D 90
What does the order of operations tell you?
How many factors of 10 are in 102?
How do you find the volume of a cube?
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33
33
Possible student work: V 5 lwh V 5 2 ft • 2 ft • 2 ft; V 5 8 ft3
8 ft3 4 2 ft3 5 4
Beth should buy 4 bags of beans.
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