G8

Input/Model Expanded Form Power

(Teacher Presents)

Directions: Complete the table.
Product
22 • 24
(32)3
(2 • 3)3

Solution:

Product Expanded Form Power
22 • 24 26
(32)3 2•2•2•2•2•2 36

(2 • 3)3 • First, simplify the exponent outside the 63
parentheses.
(32)(32)(32)

• Next, simplify the exponent inside the
parentheses.
(3 • 3)(3 • 3)(3 • 3)

• First, simplify the exponent on the parentheses
(2 • 3)(2 • 3)(2 • 3)

• Use the commutative property to rearrange
numbers; keep solution in exponential notation.
23 ⋅ 33

Talking Points:
 What patterns do you see?
 What relationship do you see between the exponents in the product and the exponents in the

power?

Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE 96

Structured Guided Practice

(A/B Partners Practice)

Directions: Complete the table. Expanded Form Power
Product

(–4)2 • (–4)2

( 4 )2

(3 )4

Solution: Expanded Form Power
Product –4 • –4 • –4 • –4 (–4)4

(–4)2 • (–4)2 ( • • • )( • • • ) 8
( 4)2
(3 )(3 )(3 )(3 )= 34 4
(3 )4 3 • 3 • 3 • 3 • • • •

Talking Points:
 What patterns do you see?
 What relationship do you see between the exponents in the product and the exponents in the

power?

97 Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Complete the table. Expanded Form Power

Product Power
�21�2 • �12�3 �21�5

(73)3 79
42 • 52
(4 • 5)2
202
Solution: Expanded Form
Product �21� • �21� • �21� • �21� • �12�

�12�2 • �21�3 (73)(73)(73)
(73)3 (7 • 7 • 7)(7 • 7 • 7)(7 • 7 • 7)

(4 • 5)2 (4 • 5)(4 • 5)
4•4•5•5

Talking Points:
 What patterns do you see?
 What relationship do you see between the exponents in the product and the exponents in the
power?

Closure

Recap today’s lesson with one or more of the following questions:

MP2: What is the relationship between the exponent and the base?
MP4: What rule might apply in this situation?

Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE 98

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 1: Multiply Exponents

Objective: I will use the expanded form to explore multiplying exponents..

Vocabulary Steps:

Base: the repeated factor 1. Identify the base(s) and exponent(s).
2. Write in expanded form.
Exponent: indicates the number of times the 3. Simplify.
base is multiplied by itself; also known as 4. Check using the properties of exponents.
power

Factors: numbers that are multiplied to create
a product

Power: a product in which the factors are the
same

5 • 5 • 5 • 5  54

4 factors power
Say as, “4 copies of 5 multiplied together”
n3= 3 copies of n, multiplied together= n ⋅ n ⋅ n

Exponential Form: expressions written with
exponents

Properties of Exponents
Power to Power (an)m = anm (Multiply)
Multiply (Like Bases) am ⋅ an = am+n (Add)
Power of a Product anbn = (ab)n (Keep)

99 Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE

Homework

Unit 2 · Lesson 1: Multiply Exponents

Example # 1

Directions: Complete the table.

Product Expanded Form Power
22 • 24 26
(32)3 2•2•2•2•2•2 36

(2 • 3)3 • First, simplify the exponent outside the 63
parentheses.
(32)(32)(32)

• Next, simplify the exponent inside the
parentheses.
(3 • 3)(3 • 3)(3 • 3)

• First, simplify the exponent on the parentheses
(2 • 3)(2 • 3)(2 • 3)

• Use the commutative property to rearrange
numbers; keep solution in exponential notation.
23 ⋅ 33

Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE 100

Homework 2. (42)3
Expanded Form
Unit 2 · Lesson 1: Multiply Exponents

Directions: Complete the table.
1. 33 • 32 • 32

Expanded Form Power Power

3. ( 3 2)2 Power 4. (–5)2 • (–5) Power
Expanded Form Expanded Form

5. (52)4 Power 6. ( 2 3 4)2 Power
Expanded Form Expanded Form

101 Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE

Answer Key

Homework Power 2. (42)3 Power
Expanded Form 42⋅3 = 46
1. 33 ⋅ 32 ⋅ 32 33 + 2 + 2 → 37
Expanded Form (42 )(42 )(42 ) Power
Power (−5)2(−5)1
3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 37 3⋅2 2⋅2 (4 ⋅ 4)(4 ⋅ 4)(4 ⋅ 4)= 46
(−5)2+1
Answer: 37 x 6y 4 Answer: 46 (−5)3
4. (−5)2(−5)
3. ( 3 2)2 Power Power
Expanded Form 52⋅4 Expanded Form 2⋅2 3⋅2 4⋅2
58 (−5)2(−5)
( )( )x 3y 2 x 3y 2 4 6 8
↓↓
↓↓ (−5)(−5)(−5)

(xxxyy )(xxxyy ) (−5)3

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ Answer: (−5)3
= 6 4 6. ( 2 3 4)2

Answer: 6 4 Expanded Form
5. (52)4 ( 2 3 4)( 2 3 4)
( )( )
Expanded Form =

(52)(52)(52)(52) 4 6 8
5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 = 58
Answer: 4 6 8
Answer: 58

Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE 102

Properties of Exponents: MPs Applied MP
Multiplication
* Embedded MP
Procedural Lesson
12345678
Grade 8 · Unit 2 · Lesson 2
* * * * *
MC: 8.EE.1

Problem of the Day Student Journal Pages

Objective: I will simplify expressions using the properties of exponents. 58-63

Vocabulary Teacher Resources

Base: the repeated factor Considerations:
Continue to encourage struggling students to write the
Exponent: indicates the number of times expanded form of an exponential expression (5 4 = 5 ∙5 ∙5 ∙5).
the base is multiplied by itself; also Have students derive the rule once they understand the
known as power concept of exponential form.

Factors: numbers that are multiplied to Another option is to create a visual representation of what
create a product is occurring when using exponents. Use base ten blocks to
show students the growth of the number.

Power: a product in which the factors are the 20
same 21
22
5 • 5 • 5 • 5  54

4 factors power

Say as,“ 4 copies of 5 multiplied together” 23

n3= 3 copies of n, multiplied together = Steps:
n⋅n⋅n 1. Identify the base(s) and exponent(s).
2. Write the expression in expanded form.
Exponential Form: expressions written with 3. Simplify.
exponents

Properties of Exponents Application of MPs:
Power to Power (an)m = anm (Multiply)
Multiply (Like Bases) am ⋅ an = am+n (Add) MP4: What rule might apply in this situation?
Power of a Product anbn = (ab)n (Keep) A rule that may apply can be ______________.

MP6: How could you test your solution to see if it
answers the problem?
I can test my solution by ___________________.

103 Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE

Input/Model

(Teacher Presents)

Directions: Complete the table.

Product Expanded Form Exponential Standard
24 • 25

(34)2
(3x)2

Solution: Expanded Form Exponential Standard
Product 512
24 • 25 2 • 2 • 2 • 2 • 2 • 2• 2 • 2 • 2 29 6561
(34)2 • Simplify the exponent outside the parentheses. 38
9x²
(3x)2 (34)(34)

• Simplify the exponent inside the parentheses.

(3 • 3 • 3 • 3)(3 • 3 • 3 • 3)

• Simplify the exponent outside the parentheses.

(3x)(3x) 32x 2
• The commutative property allows you to move

your numbers around and not affect the product.

3•3•x•x

Talking Points:
 What patterns do you see?
 What relationship do you see between the exponents in the product and the exponents in the

power?

Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE 104

Structured Guided Practice

(A/B Partners Practice)

Directions: Complete the table.

Product Expanded Form Exponential Standard
–2 • (–2)6

(w 2)3

(xy)2

Solution: Expanded Form Exponential Standard
Product ─128
–2 • (–2)6 –2 • (–2) • (–2) • (–2) • (–2) • (–2) • (–2) (–2)7 w6
(w 2)3 w6
• Simplify the exponent outside the parentheses. x²y²
(xy)2 (w 2)(w 2)(w 2) x 2y 2

• Simplify the exponent inside the parentheses.
(w • w)(w • w)(w • w)

• Simplify the exponent inside the parentheses.
(x • y • x • y)

• Use the commutative property to rearrange numbers.
x•x•y•y

Talking Points:
 What patterns do you see?
 What relationship do you see between the exponents in the product and the exponents in the

power?

105 Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE

Final Check for Understanding

(Teacher Checks Work)
Directions: Complete the table.

Product Expanded Form Exponential Standard
x3•x4

(32)3

(4y)5

Solution: Expanded Form Exponential Standard
Product x7 x7
x3•x4 x•x•x•x•x•x•x
(32)3 • Simplify the exponent outside the parentheses. 729
36
(4y)5 (32)(32)(32)
• Simplify the exponent inside the parentheses. 45y 5 1024y 5

(3 • 3)(3 • 3)(3 • 3)
• Simplify the exponent outside the parentheses.

(4 • y)(4 • y)(4 • y)(4 • y)(4 • y)
• Use the commutative property to rearrange.

4•4•4•4•4•y•y•y•y•y

Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE 106

Student Practice Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 2: Properties of Exponents:
Multiplication

Directions: Simplify. Write solution in exponential form and standard form.

1. 33 • 34 2. (33)4

Solution: Solution:

33 + 4 33 • 4
Exponential form  37 Exponential form  312
Standard form  2,187 Standard form  531,441

3. (2x)4 4. (–4)2 • (–4)3

Solution: Solution:
(2)4 (x)4 (–4)2 + 3
Exponential form: 24x 4 Exponential form  (-4)5
Standard form: 16x 4 Standard form  –1,024

5. �32�2 ⋅ �23�3 6. (m 2m 3)2

Solution: Solution:
�2�2+3 (m 2)2 (m 3)2
3 m 4m 6
�32�5 m4+6
Exponential form  Exponential form: m 10
Standard form: m 10
Standard form  32
243 Grade 8 Unit 2 Lesson 2 P TE

107 Copyright © Swun Math

Challenge Problems

Directions: Solve. 2. Describe and correct the error.
1. What is the difference between the two ( 3)7
expressions below? ( )3+7
10
–22 (–2)2

Solution: Solution:
• –22 can be expressed as –1 • 22, therefore the base • ( 3)7 indicates ( 3)( 3)( 3)( 3)( 3)( 3)( 3)
• The exponents should be multiplied not added.
is 2 and not –2. In this case, the solution is –4. • The correct solution should be 21
• Whereas, in (–2)2 the negative is included in the

parentheses, making the base –2; in this case, the

expression needs to be written as –2 • –2 with a

solution of positive 4.

Answer: –22 = –4 verses (–2)2 = 4

Extension Activity

* MP1: Make sense of the problem and persevere in solving it.

* MP4: Apply mathematics in everyday life.

Use the numbers 1-6 to make each equation true. Be sure to use each number only once.

 ·  = 
( )  = 

Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE 108

Closure

Reaching Consensus
*MP3: Do you agree or disagree with your classmate? Why or why not?

Student Presentations
*MP1: What steps in the process are you most confident about?
*MP6: Explain how you might show that your solution answers the problem.

Closure

Recap today’s lesson with one or more of the following questions:
MP4: What rule might apply in this situation?
MP6: How could you test your solution to see if it answers the problem?

109 Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 2: Properties of Exponents:
Multiplication

Objective: I will use the properties of exponents when multiplying.

Vocabulary Steps:

Base: the repeated factor 1. Identify the base(s) and exponent(s).
2. Write the expression in expanded form.
Exponent: indicates the number of times the 3. Simplify.
base is multiplied by itself; also known as
power

Factors: numbers that are multiplied to create
a product

Power: a product in which the factors are the
same

5 • 5 • 5 • 5  54

4 factors power

Say as-“ 4 copies of 5 multiplied together”
n3= 3 copies of n, multiplied together= n ⋅ n ⋅ n

Exponential Form: expressions written with
exponents

Properties of Exponents Grade 8 Unit 2 Lesson 2 P TE 110
Power to Power (an)m = anm (Multiply)
Multiply (Like Bases) am ⋅ an = am+n (Add)
Power of a Product anbn = (ab)n (Keep)

Copyright © Swun Math

Homework

Unit 2 · Lesson 2: Properties of Exponents:
Multiplication

Example
Directions: Complete the table.

Product Expanded Form Exponential Standard
24 • 25 29 512
(34)2 2 • 2 • 2 • 2 • 2 • 2• 2 • 2 • 2
6561
(3x)2 • First, simplify the exponent outside the 38
parentheses.
(34)(34) 32x 2 9x²

• Next, simplify the exponent inside the or
parentheses. 9x 2
(3 • 3 • 3 • 3)(3 • 3 • 3 • 3)

• First, simplify the exponent outside the
parentheses.
(3x)(3x)

• Use the commutative property to rearrange
numbers.
3•3•x•x

111 Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE

Homework

Unit 2 · Lesson 2: Properties of Exponents:
Multiplication

Directions: Simplify. Write solution in exponential and standard form.

1. 72 • 73 2. (3m)2

3. (–2)3(–2)3 4. (x 2y)2

5. (x 3)7 6. ((5)2)3

Explain the steps you used to solve problem number _______.

______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________

Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE 112

Answer Key 2. 32m 2
or
Extension Activity (3m)(3m)
3 • 3 • m • m = 9m 2
Answers will vary. Sample response: Answer: 32m 2 or 9m 2

1 ⋅ 4 = 5 4. x 2 • 2 y 2
( 2)3 = 6 x4 •y2

Homework Answer: x 4 • y 2 or x 4 y 2

1. + = 6. (5)(5) = 5² so then
or ( ) =
7 • 7 • 7 • 7 • 7 = 75 or
Answer: 75 or 16,807 (5)2 • 3
(5)6
3. (–2)3 + 3
(–2)6 Answer: (5)6 or 15,625
or
(–2) (–2) (–2) (–2) (–2) (–2)
(–2)6
Answer: (–2)6 or 64

5. x 3 • 7
x 21
Answer: x 21

113 Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE

Notes

Copyright © Swun Math Grade 8 Unit 2 Lesson 2 P TE 114

Divide Exponents MPs Applied MP 8
* Embedded MP
Conceptual Lesson 123 *
Grade 8 · Unit 2 · Lesson 3 4567
*
MC: 8.EE.1 

Problem of the Day Student Journal Pages

Objective: I will use expanded form to show division with exponents. 64-67

Vocabulary Teacher Resources

Properties of Exponents Considerations:

Power to Power (an)m = anm (Multiply) Have students write the expanded form of an
exponential expression. Students should practice
Multiply (Like Bases) am ⋅ an = am+n (Add) expanding and canceling out (copies). Practice paying
attention to where the extra “copies” of terms are, and
Power of a Product anbn = (ab)n (Keep) how many. This will help support the visual
understanding of the properties.

Division (Like Bases) an = an−m (Subtract) Students can evaluate the original expression to see
am the relationship between standard and exponential
form:

25 = 23 = 8 or 32 = 8
22 4

Steps:

1. Identify the base(s) and exponent(s).

2. Write the exponential notation in expanded

form.

3. Cancel all fractions that are equivalent to 1.
3
3 = 1 and = 1

4. Simplify.

5. Check using the properties of exponents.

Application of MPs:

MP2: What is the relationship between the exponent
in the division problem and the quotient?
The exponents in the quotient represent the
__________________________________________.

MP4: Where are exponents used outside of school?
Outside of school, exponents are used in
_______________________________.

115 Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE

Input/Model

(Teacher Presents)

Directions: Simplify. Write solution in standard form.

1. 25
22

Solution:

Expanded Form: Using Properties:
2⋅2⋅2⋅2⋅2 25
• 2∙2 • 22 = 25−2

• 23 = 8

• 2 ⋅ 2 ⋅ 2⋅2⋅2 = 23
2 2 1

•

• 2 ⋅ 2 ⋅ 2⋅2⋅2 = 23
2 2 1

• 23 = 8

Teacher Talking Points:

• How many factors are in the numerator?

• How many factors are in the denominator?

• How many common factors are there between the denominator and the numerator?
2
• Remember a common factor in the numerator and the denominator such as 2 is equivalent to 1.

• How many factors are left once you removed the common factors?

2. (−3)4⋅ (−3)3
(−3)5

Solution: Using Properties: (−3)7
(−3)4∙(−3)3 (−3)4+3 (−3)5
Expanded Form: • (−3)5 = (−3)5 =
• (−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3)

(−3)∙(−3)∙(−3)∙(−3)∙(−3)

• (−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3) → (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3)∙(−3) • (−3)7 = (−3)7−5
(−3)∙(−3)∙(−3)∙(−3)∙(−3) (−3) (−3) (−3) (−3) (−3) 1 (−3)5

• (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3)∙(−3) • (−3)2 = 9
(−3) (−3) (−3) (−3) (−3) 1

• (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3)∙(−3) = (−3)2
(−3) (−3) (−3) (−3) (−3) 1

• (−3)2 = 9

Important: There must be parentheses around the (–3) to have the meaning of –3 • –3 • –3. If there are no parentheses, –33,
means –3 • 3 • 3 (only one negative sign). Although the value is the same in this problem, the clarification needs to be made
clear. If the exponent is an even number, the value will not be the same.

Teacher Talking Points:

• How many factors are in the numerator?

• How many factors are in the denominator?

• How many common factors are there between the denominator and the numerator?
−3
• Remember a common factor in the numerator and the denominator such as −3 is equivalent to 1.

• How many factors are left once you removed the common factors?

Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE 116

Structured Guided Practice

(A/B Partners Practice)

Directions: Simplify. Write solution in standard form.
�−42� 3

1. −4

Solution:
-4z2

73 4
2. 73 2

Solution:
a2

117 Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Simplify. Write solution in standard form.

1. (− )10
(− )6

Solution:
(− )4

2. 4 2
2

Solution:

x3

Closure

Recap today’s lesson with one or more of the following questions:

MP2:What is the relationship between the exponent in the division problem and the
quotient?

MP4:Where are exponents used outside of school?

Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE 118

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 3: Divide Exponents

Objective: I will use the expanded form to show the division of exponents.

Vocabulary Steps:

Properties of Exponents 1. Identify the base(s) and exponent(s).

Power to Power (an)m = anm (Multiply) 2. Write the exponential notation in expanded

form.

Multiply (Like Bases) am ⋅ an = am+n (Add) 3. Cancel all fractions that are equivalent to 1.
3
3 = 1 and = 1

Power of a Product anbn = (ab)n (Keep) 4. Simplify.

5. Check using the properties of exponents.

Division (Like Bases) an = an−m (Subtract)
am

Example # 1
Directions: Simplify. Write solution in standard form

25
22

Solution:

Expanded Form: Property:
2⋅2⋅2⋅2⋅2 25
• 2∙2 • 22 = 25−2

• 23 = 8

• 2 ⋅ 2 ⋅ 2⋅2⋅2 = 23
2 2 1

•

• 2 ⋅ 2 ⋅ 2⋅2⋅2 = 23
2 2 1

• 23 = 8

Example # 2 (−3)4 ⋅ (−3)3
(−3)5
Solution:
Expanded Form: Property: (−3)7
• (−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3) (−3)4∙(−3)3 (−3)4+3 (−3)5
• (−3)5 = (−3)5 =
(−3)∙(−3)∙(−3)∙(−3)∙(−3)

• (−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3)∙(−3) → (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3)∙(−3) • (−3)7 = (−3)7−5
(−3)∙(−3)∙(−3)∙(−3)∙(−3) (−3) (−3) (−3) (−3) (−3) 1 (−3)5

• (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3)∙(−3) • (−3)2 = 9
(−3) (−3) (−3) (−3) (−3) 1

• (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3)∙(−3) = (−3)2
(−3) (−3) (−3) (−3) (−3) 1

• (−3)2 = 9

119 Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE

Homework

Unit 2 · Lesson 3: Divide Exponents

Directions: Simplify. Write solution in standard form.

37 6
1. 34 2. 2

(−5)2⋅(−5)2 10 8
3. −5 4. 2 3

(4.2)6 4 9 8
5. (4.2)5 4 6.

Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE 120

Answer Key

Homework

1. 37−4 2. x 6 – 2
33 = 27 x4

3. (−5)2+2 4. x10 ⋅ y8 = x10−2 ⋅ y8−3
x2 y3
(−−55)4

(−−551)4 (−5)4−1 x8y5
−51
=

(−5)3 = −125

5. (4.2)6 ⋅ 4 6. 9 8
(4.2)5 4

(4.2)6−5 ⋅ 4−4 Remember anything to the power 9 8
(4.2)1 0 = 4.2 of 0 is equivalent to 1. ′ 1

9−1 8−1= 8 7

121 Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE

Notes

Copyright © Swun Math Grade 8 Unit 2 Lesson 3 C TE 122

Properties of Exponents: Division MPs Applied MP

Procedural Lesson * Embedded MP

Grade 8 · Unit 2 · Lesson 4 1234567 8

MC: 8.EE.1 ** * * *

Problem of the Day Student Journal Pages

68-73

Objective: I will simplify expressions using the properties of exponents when dividing.

Vocabulary Teacher Resources

Properties of Exponents Considerations:

Power to Power (an)m = anm (Multiply) Encourage the use of expanded notation if using the property
is too difficult. Students should be able to derive the rule once
Multiply (Like Bases) am ⋅ an = am+n (Add) they understand the concept in exponential form. To simplify
an expression involving monomials, write an equivalent
Power of a Product anbn = (ab)n (Keep) expression in which:

Division (Like Bases) an = an−m (Subtract) • Each base appears exactly once
am
• There are no powers of powers

• Fractions are in simplest form

Steps:

1. Identify the base(s) and exponent(s).

2. Write in expanded form.

3. Cancel all fractions that are equivalent to 1.
3
3 = 1 and = 1

4. Simplify.

5. Check using the properties of exponents.

Application of MPs:

MP2: What is the relationship between the exponent in the
division problem and the quotient?
The exponents in the quotient represent the
__________________________________________.

MP6: When must there be parentheses in the answer?
The parentheses must be included in the answer
because ____________________________.

123 Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE

Input/Model

(Teacher Presents)

Directions: Simplify. Write answer in standard form.

4⋅ 3 3⋅ 4⋅ 3
1. 6 2. 2⋅

Solution: Solution:
3⋅ 4 ⋅ 3
• 4 ⋅ 3 = 4+3 • 2⋅ 1 = 3−2 4−1 3
6 6
7
• 6 = 7−6 • 1 ⋅ 3 ⋅ 3

• 1

• 3 3

2 5
3. 6 3

Solution: 5−3
2−1 1
• 1 ⋅ 1 ⋅
6
2 1 2
• 1 ⋅ ⋅ 1 = 6
6 1
2
• 6

Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE 124

Structured Guided Practice

(A/B Partners Practice)

Directions: Simplify. Write answer in standard form.

68 10⋅ 8⋅ 6
1. 65⋅6 2. 2⋅ 7

Solution: Solution:
62 36 x8yz6

54 Grade 8 Unit 2 Lesson 4 P TE
3. 32⋅52

Solution:
25
9

125 Copyright © Swun Math

Final Check for Understanding

(Teacher Checks Work)

Directions: Simplify. Write answer in standard form.

(−7)5⋅(−7)2 25⋅42⋅ 3
1. (−7)4 2. 22⋅

Solution: Solution:
─343 128d2

3
3. 2

Solution:
2 � �2
2

Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE 126

Student Practice Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 4: Properties of Exponents: Division

Directions: Simplify. Write answer in standard form.

1. 1110 2. 82⋅83
118 2 82

Solution: Solution:
121 83 = 512
2
4. 152⋅ 5
10 2

3. 2⋅ 3 Solution:
225d3
Solution:
7
2

5. 12 6. 4⋅ 3
5⋅ 3

Solution: Solution:
m4 3 3 ( )3

127 Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE

Challenge Problems

Directions: Solve.

1. Describe and correct the error. 2. Simplify. 8 ·
─3
615 = 615÷3 = 65
63

Solution: Solution:
Even though the expressions are being divided, the 8 · = 9
exponents should not be divided.
The exponents should be subtracted. 9 9 3
The correct solution should be: 615−3 = 612 −3 ·1

9 · 3 = 12

Extension Activity

* MP1: Make sense of the problem and persevere in solving it.

* MP4: Apply mathematics in everyday life.

Make the equation below true using only the numbers 1-8. Be sure to use each number only
once.

 ·  ·  =  
 ·  · 

Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE 128

Closure

Reaching Consensus
*MP3: Do you agree or disagree with your classmate? Why or why not?

Student Presentations
*MP1: What steps in the process are you most confident about?
*MP6: Explain how you might show that your solution answers the problem.

Closure

Recap today’s lesson with one or more of the following questions:
MP2: What is the relationship between the exponent in the division problem and

the quotient?
MP6: When must there be parentheses in the answer?

129 Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 4: Properties of Exponents: Division

Objective: I will simplify expressions using the properties of exponents when dividing.

Vocabulary Steps:

Properties of Exponents 1. Identify the base(s) and exponent(s).
Power to Power (an)m = anm (Multiply)
2. Write the exponential notation in expanded

Multiply (Like Bases) am ⋅ an = am+n (Add) form.

3. Cancel all fractions that are equivalent to 1.
3
Power of a Product anbn = (ab)n (Keep) 3 = 1 and =1

an an−m 4. Simplify.
am
Division (Like Bases) = (Subtract) 5. Check using the properties of exponents.

Example # 1 Example # 2

Directions: Simplify. Write answer in standard form.

4 ⋅ 3 Solution: 3 ⋅ 4 ⋅ 3
6 2 ⋅

Solution: 3⋅ 4⋅ 3
4 ⋅ 3 4+3 2⋅ 1
• 6 = 6 • = 3−2 4−1 3

• 7 = 7−6 • 1 ⋅ 3 ⋅ 3
6 • 3 3
• 1

Example # 3

Solution: 2 5
6 3

• 1 ⋅ 1 21⋅− 1 1 2⋅ 5−3
• 6 ⋅
1 =1 16 2
6
2
• 6

Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE 130

Homework

Unit 2 · Lesson 4: Properties of Exponents: Division

Directions: Simplify. Write answer in standard form.

1. 89 2. 75⋅73
87 72

3. 22 4. 5
10⋅ 5 63⋅ 2

5. 410 3 6. 2 3 9
410 2 5

Explain the steps you used to solve problem number _______.

______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________

131 Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE

Answer Key

Extension Activity

Answers will vary. Sample responses:

8 ⋅ 4 ⋅ 6 = 5 ⋅ 3
7 ⋅ 1 ⋅ 2

6 ⋅ 5 ⋅ 7 = 3 ⋅ 4
8 ⋅ 1 ⋅ 2

Homework
89
1. 87 = 89−7 2. = ⋅



82 = 64 78−2 = 76

76 = 117,649

3. = 4. = ⋅
+
′ ⋅ ⋅

22 = 22−15 1 ⋅ 5−2
15 63 1

7 1 ⋅ 3
63 1

3 = � �3 3
63 6 216

5. 410 3 = 410 3 6. 2 3 9 = 2 ⋅ 3−2 ⋅ 9−5
410 410 1 2 5

410−10 3−1 2 ⋅ 1 ⋅ 4 2 4

40 2 = 2

Copyright © Swun Math Grade 8 Unit 2 Lesson 4 P TE 132

Properties of Exponents: MPs Applied MP
Zero and Negatives
* Embedded MP
Conceptual Lesson
12 3 4567 8
Grade 8 · Unit 2 · Lesson 5
* *
MC: 8.EE.1

Problem of the Day Student Journal Pages

Objective: I will explore patterns with positive and negative exponents. 74-77

Vocabulary Teacher Resources

Negative Exponent: indicates the number of Considerations:
times to divide by the base
Create a chart like the one being created today to post on
−2 the wall. Students should explore the patterns found in
the chart. Allow time for discussion on the use of
The negative sign says to do the inverse. multiplication when there is a positive exponent and
Dividing is the inverse of multiplying. division when there is a negative exponent.

It is critical for students to understand that decreasing
an exponent by one is the same as dividing by the base.
Use other bases to construct the same chart if students
continue to struggle.

Reciprocal: what to multiply a value by to get 1;

multiplicative inverse Steps:

A negative exponent is equivalent to the inverse of 1. Complete the chart.
the positive of the same number. 2. Use expanded form to identify the standard

x−3 is the reciprocal of x3 notation.
3. Identify the pattern.
4−5 is the reciprocal of 45 4. Use the pattern to identify the standard notation.

Properties of Exponents

Zero Exponent a0 = 1 Application of MPs:

MP3: Why is any number to the zero-power equal to 1?

Negative Exponent a−n = 1 Any number to the zero power is equal to 1
an
because .

MP8: What patterns do you notice? .
The patterns I notice are

133 Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE

Input/Model

(Teacher Presents)

Directions: Write the expanded and standard form.

1. 2.
Power
24 Expanded Standard Power Expanded Standard
23 23 2⋅2⋅2 8
22 2⋅2 4
21 22 2
20 2
21

20

2−1

2−2

2−3

Solution: Solution:
• Begin by recording the expanded form and standard • Continue to build the base 2 chart.

form for the powers of 24, 23, 22, and 21 (not 20). • Review the pattern found in input 1. Remind students

• Discuss the decreasing pattern in the standard form that each value is divided by the “base,” in this case 2.

column. Each value decreases by a factor of 2, which is • Note the chart is partially filled in; continue building
the chart beginning with 20 (to reinforce that 20 = 1).
equivalent to dividing by two.
• Have students predict what the value of 20 is equivalent • Using the established pattern, determine that since
20 = 1, 1 will be divided by the base 2 to yield the value of
to. 2-1.

• Using the pattern established, determine that since • 1 ÷ 2 =1 . Therefore, 2−1 = 1
21 = 2, 2 will be divided by the base to yield the value of 2 2
20.
• Continue the pattern to find 2−2 and 2−3.
• 2 ÷ 2 = 1; Therefore, 20 = 1
1 =1 2−2 1
• 2 ÷ 2 4 . Therefore = 4

Power Expanded Standard • 1 ÷ 2 =1 . Therefore 2−3 = 1
24 2⋅2⋅2⋅2 4 8 8
23 16
22 2⋅2⋅2 • Note: students may need to be reminded how to divide
21 2⋅2 8
20 2 fractions by a whole number.
4
1 2 Power Expanded Standard 1 ÷2
1 23 2⋅2⋅2
22 2⋅2 8 1
21 2 ÷2
20 2 4 1
2 4 ÷2
2−1 1 1

2−2 1 1
2 2
2−3
1 ⋅ 1 1
2 2 4

1 ⋅ 1 ⋅ 1 1
2 2 2 8

Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE 134

Structured Guided Practice

(A/B Partners Practice)

Directions: Write the expanded and standard form.

1. 2.

Power Expanded Standard Power Expanded Standard
34 33 3•3•3 27
33 32 9
32 31 3•3 3
31 30 3
30 3–1
3–2
3–3

Solution: Solution:

Power Expanded Standard Power Expanded Standard
34 3•3•3•3 81 33 3•3•3 27
27
33 3 • 3 • 3 9 81 ÷ 3 32 3 • 3 9
3 27 ÷ 3
32 3 • 3 1
9 ÷3
31 3 3÷3 31 3 3
1
30 1
30 1
1 ÷3
3–1 1 1 1
3 3 3 ÷3
1
3–2 11 1 or 1 9 ÷3
3• 3 9 32

3–3 111 1 or 1
3• 3• 3 27 33

135 Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Write the expanded and standard form.

1. Expanded Standard 2. Expanded Standard

Power Power
54 53
53 52
52 51
51 50
50 5–1
5–2
5–3

Solution: Solution:

Power Expanded Standard Power Expanded Standard
54 5•5•5•5 625 53 5•5•5 125
125
53 5 • 5 • 5 25 52 5 • 5
5
1 25

52 5 • 5

51 5 51 5 5

50 1 50 1 1
1
1
5–1 5 5

5–2 11 1 or 1
5•5 25 52

5–3 111 1 or 1
5•5•5 125 53

Closure

Recap today’s lesson with one or more of the following questions:

MP3: Why is any number to the zero-power equal to 1?
MP8: What patterns do you notice?

Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE 136

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 5: Properties of Exponents:
Zero & Negatives

Objective: I will explore patterns with positive and negative exponents.

Vocabulary Steps:

Negative Exponent: indicates the number of 1. Complete the chart.
times to divide by the base 2. Use expanded form to identify the standard

−2 notation.
The negative sign says to do the inverse. 3. Identify the pattern.
4. Use the pattern to identify the standard notation.
Dividing is the inverse of multiplying.

Reciprocal: what to multiply a value by to get
1; multiplicative inverse

A negative exponent is equivalent to the inverse
of the positive of the same number.

x−3 is the reciprocal of x3

4−5 is the reciprocal of 45

Properties of Exponents

Zero Exponent a0 = 1 1
an
Negative Exponent a−n =

Example # 1 Example # 2

Directions: Use pattern reasoning to find the expanded and standard form.

Solution: Solution:
• Using the pattern established, determine that since 21 =
• 1 ÷ 2 =1 . Therefore, 2−1 = 1
2, 2 will be divided by the base to yield the value of 20. 2 2

• 2 ÷ 2 = 1. Therefore, 20 = 1 • Continue the pattern to find 2−2 and 2−3.

• 1 ÷ 2 =1 . Therefore 2−2 = 1
2 4 4
Power Expanded Form Standard Form
24 2⋅2⋅2⋅2 16 • 1 ÷ 2 =1 . Therefore 2−3 = 1
4 8 8

23 2 ⋅ 2 ⋅ 2 8 Power Expanded Form Standard Form
23 2⋅2⋅2 8
22 2 ⋅ 2 4 22 2⋅2 4
21 2 2
21 2 2 20 1 1 1 ÷2

2−1 1 1 1
2 2 ÷2
20 1 1 2 1
4 ÷2
2−2 1⋅1 1
22 4

2−3 1 ⋅ 1 ⋅ 1 1
222 8

137 Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE

Homework

Unit 2 · Lesson 5: Properties of Exponents:
Zero & Negatives

Directions: Use the expanded form to find the standard form.

1. Complete the chart. 2. 6−2

Power Expanded Standard
43
42
41
40
4−1
4−2
4−3

3. 60 4. 7−2

5. 6−3 6. 110

Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE 138

Answer Key

Homework 2. 1
36
1. Expanded Standard
Power 4⋅4⋅4 64
43
42 4⋅4 16
41
40 4 4
4−1
1 1
4−2
1 1
4−3 4 4

1 ⋅ 1 1 1
4 4 16 42

1 ⋅ 1 ⋅ 1 1 1
4 4 4 64 43

3. 1 4. 1
49

5. 1 6. 1
216

139 Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE

Notes

Copyright © Swun Math Grade 8 Unit 2 Lesson 5 C TE 140

Convert Negative Exponents MPs Applied MP

Conceptual Lesson * Embedded MP
Grade 8 · Unit 2 · Lesson 6
12345678
MC: 8.EE.1
*  *

Problem of the Day Student Journal Pages

78-81

Objective: I will use the reciprocal to convert negative exponents to positive.

Vocabulary Teacher Resources
Negative Exponent: indicates the number of
times to divide by the base Considerations:
−2
Review the chart from yesterday’s lesson to remind
The negative sign says to do the inverse. students of the patterns between negative and
Dividing is the inverse of multiplying. positive exponents. Allow time for discussion on the
use of multiplication when there is a positive
Reciprocal: what to multiply a value by to get 1; exponent and division when there is a negative
multiplicative inverse exponent (inverse relationship).

A negative exponent is equivalent to the inverse of the Review the concept that 52 and 5–2 are reciprocals.
positive of the same number.
• 52 means multiply 5 twice. 52 = 5 x 5 = 25
x−3 is the reciprocal of x3 • 5-2 means to do the exact opposite operation. So,
4−5 is the reciprocal of 45
instead of multiplying, divide.

5-2 = 1 ⋅ 1 = 1 = 1
5 5 25 52

Steps:
1. Identify the negative exponent.
2. Write the reciprocal.
3. Simplify.

Properties of Exponents

Negative Exponent a−n = 1 Application of MPs:
an
MP1: How are negative exponents and reciprocals

related?

Negative exponents and reciprocals are

related because .

MP7: What is the relationship between x y and x −y ?

The relationship between x y and x −y is that

they .

141 Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE

Input/Model

(Teacher Presents)

Directions: Write each expression using positive exponents.

1. 2−3 2. 1
2−3

Solution: 1
2−3
• Identify that the denominator can be rewritten

Solution: using its inverse. 1
1
Use the patterns from yesterday’s chart or today’s 23

review of reciprocals. • The fraction bar also represents a division.

• 2−1 is equivalent to 1 Rewrite as a division problem.
2
1 1 1 1 11
• 2−2is equivalent to 2 ⋅ 2 = 4 = 22 = 1 ÷ 23
1
• Therefore 2−3 is equivalent to 1 ⋅ 1 ⋅ 1 = 1 = 1 23
2 2 2 8 23
1 • Lastly, simplify by multiplying by its reciprocal:
23
Answer: 1 · 23

Answer: 23

Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE 142

Structured Guided Practice

(A/B Partners Practice)

Directions: Write each expression using positive exponents.

1. 4−2 2. 1
4−2

Solution:

Solution: 1 = 42
4−2
4−2 = 1
42

143 Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Write each expression using positive exponents.

1. 5−4 2. 1
5−4

Solution: Solution:

5−4 = 1 1 = 54
54 5−4

Closure

Recap today’s lesson with one or more of the following questions:

MP1: How are negative exponents and reciprocals related?
MP7: What is the relationship between and − ?

Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE 144

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 6: Convert Negative Exponents

Objective: I will use the reciprocal to convert negative exponents to positives.

Vocabulary Steps:

Negative Exponent: indicates the number of 1. Identify the negative exponent.
times to divide by the base 2. Write the reciprocal.
3. Simplify.
−2

The negative sign says to do the inverse.
Dividing is the inverse of multiplying.

Reciprocal: what to multiply a value by to get 1;
multiplicative inverse

A negative exponent is equivalent to the inverse of the
positive of the same number.

x−3 is the reciprocal of x3

4−5 is the reciprocal of 45

Properties of Exponents

Negative Exponent a−n = 1
an

Example # 1 Example # 2

Directions: Write each expression using positive exponents.

2−3 1
2−3

Solution: Solution:

Use the patterns from yesterday’s chart or today’s • Identify that the denominator 1 can be rewritten using its
2−3
review of reciprocals. 1 1
2
• 2−1 is equivalent to inverse. 1
23
2−2is equivalent to 1 ⋅ 1 1 1
• 2 2 = 4 = 22 • The fraction bar also represents a division. Rewrite as a

• Therefore 2−3 is equivalent to 1 ⋅ 1 ⋅ 1 = 1 = 1 division problem.
2 2 2 8 23 11
1 1 = 1 ÷ 23
Answer: 23
23

• Lastly, simplify by multiplying by its reciprocal:

1 · 23

Answer: 23

145 Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE