Name ____________________________ Period ____ Math Number ____
1
The following CCSS is addressed in : G8, M1, TA, L1-6
8 EE 1
Know and apply the properties of integer exponents to generate equivalent
numerical expressions.
Lesson 1: Exponential Notation .......................................................pages 2 - 9
a. Students know what it means for a number to be raised to a power and how to represent
the repeated multiplication symbolically.
b. Students know the reason for some bases requiring parentheses.
Lesson 2: Multiplication of Numbers in Exponential Form...pages 10 - 15
c. Students use the definition of exponential notation to make sense of the first law of
exponents.
d. Students see a rule for simplifying exponential expressions involving division as a
consequence of the first law of exponents.
e. Students write equivalent numerical and symbolic expressions using the first law
of exponents.
Lesson 3: Numbers in Exponential Form Raised to a Power..pages 16 - 18
f. Students will know how to take powers of powers. Students will know that when a
product is raised to a power, each factor of the product is raised to that power.
g. Students will write simplified, equivalent numeric and symbolic expressions using this
new knowledge of powers.
Lesson 4: Numbers Raised to the Zeroth Power .........................pages 19 - 21
h. Students know that a number raised to the zeroth power is equal to one.
i. Students recognize the need for the definition to preserve the properties of exponents.
Lesson 5: Negative Exponents and the Laws of Exponents ... pages 22 - 27
j. Students know the definition of a number raised to a negative exponent.
k. Students simplify and write equivalent expressions that contain negative exponents.
Lesson 6: Proofs of Laws of Exponents ................................................pages 28 - 30
l. Students extend the previous laws of exponents to include all integer exponents.
m. Students base symbolic proofs on concrete examples to show that (xb)a = xab is valid for
all integer exponents.
Mixed Practice .......................................................................................pages 31 - 37
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
2
LESSON 1: Exponential Notation
MATH FACT: The equal sign (=) was invented by mathematician Robert Recorde in 1557.
1. Compare: expression, an equation, and an inequality
2. Suppose you have the EXPRESSION 52 , then the “5” is called the
___________________ and the “2” is called the ___________________ or
___________________. The expression 52 means to multiply 5 by itself 2 times. It would
look like this: 5 x 5 = 25. This expression is read five squared or five to the second
power.
3. Suppose you have the expression 4 6 then how would you write this as a multiplication
problem? ____________________________________________________ and to what
would it be equal? ___________________.
SHOW WORK HERE:
What do our special terms "squared" and "cubed" look like? Draw a picture of 𝟒𝟐
and of 𝟒𝟑 .
Length
Area
Volume
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
3
4. It is your lucky day, your favorite relative is going to give you cash!!! You have to decide
if you would rather have $ 53 or $ 35 . Explain which you choose and explain why.
5. It is your lucky day, your favorite relative is going to give you cash!!! You have to decide
if you would rather have $ 2 4 or $ 4 2 . Explain which you choose and explain why.
ACTIVITY:
Students may work individually or in a small group to answer each question. Remember,
you must have "proof" for your answer, not just an answer.
1. When a negative number is raised to an odd power, what is the sign of the result?
2. When a negative number is raised to an even power, what is the sign of the result?
CONCLUSION: When a negative number is raised to an odd power, the sign of the answer
is __________. Conversely, if a negative number is raised to an even power, the sign of the
answer is __________.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
4
Now, suppose you have an ENTIRE fraction being raised to a power…
But, what would it look like if the entire fraction wasn’t raised to a power, but only the
numerator or the denominator?
3

Using EXPONENTS is a shortcut way to represent repeated multiplication of the
same number.
7
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
5
When you apply an exponent to a base, you have to be very careful when the base is a
negative number because the exponent only applies to the base.
Example 1:
−53 = (−1)(53 ) = (−1)(5)(5)(5) = −𝟏𝟐𝟓
𝑤ℎ𝑒𝑟𝑒𝑎𝑠
(−5)3 = (−5)(−5)(−5) = −𝟏𝟐𝟓
Example 2:
−52 = (−1)(52 ) = (−1)(5)(5) = −𝟐𝟓
𝑤ℎ𝑒𝑟𝑒𝑎𝑠
(−5)2 = (−5)(−5) = 𝟐𝟓
Let’s try some:
( 3
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
Make up at least three (3) problems that show what happens when a negative number is
raised to an odd power. Is the sign of the answer positive or negative?
Ex 1:
Ex 2:
Ex 3:
Extra:
Make up at least three (3) problems that show what happens when a negative number is
raised to an even power. Is the sign of the answer positive or negative?
Ex 1:
Ex 2:
Ex 3:
Extra:
What pattern(s) do you notice?
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
6
Name ____________________________ Period ____ Math Number ____
Exercise 1
Exercise 6
4 × ⋯× 4 =
⏟
7
7
× ⋯× =
⏟
2
2
7 times
21 times
Exercise 2
Exercise 7
3.6 × ⋯ × 3.6 = 3.647
⏟
(−13) × ⋯ × (−13) =
⏟
_______ times
6 times
Exercise 3
Exercise 8
(−11.63) × ⋯ × (−11.63) =
⏟
1
1
(− ) × ⋯ × (− ) =
⏟ 14
14
34 times
10 times
Exercise 4
Exercise 9
12 × ⋯ × 12 = 1215
⏟
𝑥 ∙ 𝑥⋯𝑥 =
⏟
185 times
_______times
Exercise 5
Exercise 10
(−5) × ⋯ × (−5) =
⏟
𝑥 ∙ 𝑥 ⋯ 𝑥 = 𝑥𝑛
⏟
_______times
10 times
Exercise 11
Will these products be positive or negative? How do you know?
(−1) × (−1) × ⋯ × (−1) = (−1)12
⏟
12 times
(−1)
× (−1) × ⋯ × (−1) = (−1)13
⏟
13 times
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
7
Name ____________________________ Period ____ Math Number ____
Exercise 12
Is it necessary to do all of the calculations to determine the sign of the product? Why or why not?
(−5) × (−5) × ⋯ × (−5) = (−5)95
⏟
95 times
(−1.8) × (−1.8) × ⋯ × (−1.8) = (−1.8)122
⏟
122 times
Exercise 13
Fill in the blanks about whether the number is positive or negative.
If 𝑛 is a positive even number, then (−55)𝑛 is __________________________.
If 𝑛 is a positive odd number, then (−72.4)𝑛 is __________________________.
Exercise 14
Josie says that (−15)
× ⋯ × (−15) = −156 . Is she correct? How do you know?
⏟
6 times
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
8
Name ____________________________ Period ____ Math Number ____
Problem Set
1.
Use what you know about exponential notation to complete the expressions below.
𝟑. 𝟕 × ⋯ × 𝟑. 𝟕 = 𝟑. 𝟕𝟏𝟗
⏟
(−𝟓)
× ⋯ × (−𝟓) =
⏟
_____ times
𝟏𝟕 times
𝟕 × ⋯ × 𝟕 = 𝟕𝟒𝟓
⏟
𝟔×⋯×𝟔 =
⏟
𝟒 times
_____ times
(−𝟏.
⏟ 𝟏) × ⋯ × (−𝟏. 𝟏) =
𝟒. 𝟑 × ⋯ × 𝟒. 𝟑 =
⏟
𝟏𝟑 times
𝟗 times
𝟏𝟏
𝟏𝟏
𝟏𝟏 𝒙
(− ) × ⋯ × (− ) = (− )
⏟ 𝟓
𝟓
𝟓
𝟐
𝟐
( ) × ⋯× ( ) =
⏟𝟑
𝟑
𝟏𝟗 times
_____ times
(−𝟏𝟐) × ⋯ × (−𝟏𝟐) = (−𝟏𝟐)𝟏𝟓
⏟
𝒂×⋯×𝒂 =
⏟
𝒎 times
_____ times
2.
Write an expression with (−1) as its base that will produce a positive product.
3.
Write an expression with (−1) as its base that will produce a negative product.
4.
Rewrite each number in exponential notation using 2 as the base.
8=
64 =
16 =
128 =
32 =
256 =
5.
Tim wrote 16 as (−2)4 . Is he correct?
6.
Could −2 be used as a base to rewrite 32? 64? Why or why not?
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
9
Name ____________________________ Period ____ Math Number ____
10
LESSON 2: Multiplication of Numbers in Exponential Form
First, suppose you want to multiply one expression with an exponent by another expression
with an exponent. What might you do? Let’s explore…
Sometimes the expressions have the same base:
52  54
And, sometimes the expressions do not have the same base:
52  34
52 × 53 = (5 × 5) × (5 × 5 × 5)
52 × 53 = 5 × 5 × 5 × 5 × 5
52 × 53 = 55
Simplify: 23 × 24
Simplify: 72 × 76
Simplify: 34 × 33 × 35
When two expressions with the same base are multiplied together…
I notice that…
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
Now I will show you two more
examples that follow the pattern that
was found.
Example 1:
Example 2:
I checked my classmate’s examples;
they are correct ____ incorrect ____ Initials _____
Make corrections, if needed.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
11
Name ____________________________ Period ____ Math Number ____
12
Classwork
In general, if 𝑥 is any number and 𝑚, 𝑛 are positive integers, then
𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛
because
(𝑥 ⋯ 𝑥) × ⏟
(𝑥 ⋯ 𝑥) = ⏟
(𝑥 ⋯ 𝑥) = 𝑥 𝑚+𝑛 .
𝑥𝑚 × 𝑥𝑛 = ⏟
𝑚 times
𝑛 times
𝑚+𝑛 times
Exercise 1
Exercise 5
𝟏𝟒𝟐𝟑 × 𝟏𝟒𝟖 =
Let 𝑎 be a number.
𝑎23 ∙ 𝑎8 =
Exercise 2
Exercise 6
(−72)10 × (−72)13 =
Let 𝑓 be a number.
𝑓 10 ∙ 𝑓 13 =
Exercise 3
Exercise 7
𝟓𝟗𝟒 × 𝟓𝟕𝟖 =
Let 𝑏 be a number.
𝒃𝟗𝟒 ∙ 𝒃𝟕𝟖 =
Exercise 4
Exercise 8
(−𝟑)𝟗 × (−𝟑)𝟓 =
Let 𝑥 be a positive integer. If (−3)9 × (−3) 𝑥 =
(−3)14 , what is 𝑥?
What would happen if there were more terms with the same base? Write an equivalent expression for
each problem.
Exercise 9
Exercise 10
94 × 96 × 913 =
23 × 25 × 27 × 29 =
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
13
Can the following expressions be simplified? If so, write an equivalent expression. If not, explain why not.
Exercise 11
Exercise 14
65 × 49 × 43 × 614 =
24 × 82 = 24 × 26 =
Exercise 12
Exercise 15
(−4)2 ∙ 175 ∙ (−4)3 ∙ 177 =
37 × 9 = 37 × 32 =
Exercise 13
Exercise 16
2
2
4
15 ∙ 7 ∙ 15 ∙ 7 =
54 × 211 =
Exercise 17
Let 𝑥 be a number. Simplify the expression of the following number:
(2𝑥 3 )(17𝑥 7 ) =
Exercise 18
Let 𝑎 and 𝑏 be numbers. Use the distributive law to simplify the expression of the following number:
𝑎(𝑎 + 𝑏) =
Exercise 19
Let 𝑎 and 𝑏 be numbers. Use the distributive law to simplify the expression of the following number:
𝑏(𝑎 + 𝑏) =
Exercise 20
Let 𝑎 and 𝑏 be numbers. Use the distributive law to simplify the expression of the following number:
(𝑎 + 𝑏)(𝑎 + 𝑏) =
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
14
In general, if 𝑥 is nonzero and 𝑚, 𝑛 are positive integers, then
𝑥𝑚
= 𝑥 𝑚−𝑛 , if 𝑚 > 𝑛.
𝑥𝑛
Exercise 21
Exercise 23
79
=
76
8 9
( )
5 =
8 2
( )
5
Exercise 22
Exercise 24
(−5)16
=
(−5)7
135
=
134
Exercise 25
Let 𝑎, 𝑏 be nonzero numbers. What is the following number?
𝑎 9
( )
𝑏 =
𝑎 2
( )
𝑏
Exercise 26
Let 𝑥 be a nonzero number. What is the following number?
𝑥5
=
𝑥4
Can the following expressions be simplified? If yes, write an equivalent expression for each problem. If
not, explain why not.
Exercise 27
Exercise 29
27 27
=
=
42 24
35 ∙ 28
=
32 ∙ 23
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
Exercise 28
Exercise 30
323 323
= 3 =
27
3
(−2)7 ∙ 955
=
(−2)5 ∙ 954
15
Exercise 31
Let 𝑥 be a number. Simplify the expression of each of the following numbers:
a.
b.
c.
5
𝑥3
5
𝑥3
5
𝑥3
(3𝑥 8 ) =
(−4𝑥 6 ) =
(11𝑥 4 ) =
Exercise 32
Anne used an online calculator to multiply 2,000,000,000 × 2, 000, 000, 000, 000. The answer showed
up on the calculator as 4e + 21, as shown below. Is the answer on the calculator correct? How do you
know?
.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
16
Lesson 3: Numbers in Exponential Form Raised to a Power
Class Discussion:
Classwork
For any number 𝑥 and any positive integers 𝑚 and 𝑛,
(𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛
because
(𝑥 𝑚 )𝑛 = ⏟
(𝑥 ∙ 𝑥 ⋯ 𝑥)𝑛
𝑚 times
(𝑥 ∙ 𝑥 ⋯ 𝑥) × ⋯ × ⏟
(𝑥 ∙ 𝑥 ⋯ 𝑥)
=⏟
⏟ 𝑚 times
𝑚 times
𝑛 times
= 𝑥 𝑚𝑛 .
Exercise 1
Exercise 3
(153 )9
(3.417 )4 =
=
Exercise 2
Exercise 4
((−2)5 )8 =
Let 𝑠 be a number.
(𝑠17 )4 =
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
17
For any numbers 𝑥 and 𝑦, and positive integer 𝑛,
(𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛
because
(𝑥𝑦)𝑛 = (𝑥𝑦)
⏟ ⋯ (𝑥𝑦)
𝑛 times
(𝑥 ∙ 𝑥 ⋯ 𝑥) ∙ ⏟
(𝑦 ∙ 𝑦 ⋯ 𝑦)
=⏟
𝑛 times
𝑛 times
𝑛 𝑛
=𝑥 𝑦 .
Exercise 5
Sarah wrote (35 )7 = 312 . Correct her mistake. Write an exponential expression using a base of 3 and
exponents of 5, 7, and 12 that would make her answer correct.
Exercise 6
A number 𝑦 satisfies 𝑦 24 − 256 = 0. What equation does the number 𝑥 = 𝑦 4 satisfy?
Exercise 7
Exercise 10
(11 × 4)9 =
Let 𝑥 be a number.
(5𝑥)7 =
Exercise 8
Exercise 11
(32 × 74 )5 =
Let 𝑥 and 𝑦 be numbers.
(5𝑥𝑦 2 )7 =
Exercise 9
Exercise 12
Let 𝑎, 𝑏, and 𝑐 be numbers.
Let 𝑎, 𝑏, and 𝑐 be numbers.
(32 𝑎4 )5 =
(𝑎2 𝑏𝑐 3 )4 =
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
18
Exercise 13
Let 𝑥 and 𝑦 be numbers, 𝑦 ≠ 0, and let 𝑛 be a positive integer. How is (
𝑥 𝑛
𝑦
) related to 𝑥 𝑛 and 𝑦 𝑛 ?
Problem Set
1.
Show (prove) in detail why (2 ∙ 3 ∙ 7)4 = 24 34 74 .
2.
Show (prove) in detail why (𝑥𝑦𝑧)4 = 𝑥 4 𝑦 4 𝑧 4 for any numbers 𝑥, 𝑦, 𝑧.
3.
Show (prove) in detail why (𝑥𝑦𝑧)𝑛 = 𝑥 𝑛 𝑦 𝑛 𝑧 𝑛 for any numbers , 𝑦, and 𝑧 and for any positive integer
𝑛.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
19
Lesson 4: Numbers Raised to the Zeroth Power
If you raise a number to the zero power, then what do you hypothesize its value will be?
Question
50 =?
Hypothesis
Do others in class agree with you? YES _____ NO _____ SOME _____
What, if any, are some of the other hypotheses? ____________________
Let's investigate...
In math we may often use patterns to help us find a solution. So, ...
Power
Value
35
34
33
32
31
30
Power
Value
25
24
23
22
21
20
Power
Value
15
14
13
12
11
10
What do you notice? Is there a pattern?
Let's try some together:
70 =
(−7)0 =
𝑦12
=
𝑦12
𝑥4 𝑦5
×
=
𝑦5 𝑥4
−(7)0=
23 ×
1
=
23
2.5540 =
0
7[23 𝑏] =
Connections:
8
A fraction having an equivalent numerator and denominator (for example 8 ) is equal to 1.
And
𝟑𝟒
𝟑𝟒
= 𝟑𝟒−𝟒 = 𝟑𝟎 =1
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
20
Classwork
For any numbers 𝑥, 𝑦, and any positive integers 𝑚, 𝑛, the following holds
𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛
(1)
(𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛
(2)
(𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛 .
(3)
Definition:
___________________________________________________________________________
Exercise 1
List all possible cases of whole numbers 𝑚 and 𝑛 for identity (1). More precisely, when 𝑚 > 0 and 𝑛 > 0,
we already know that (1) is correct. What are the other possible cases of 𝑚 and 𝑛 for which (1) is yet to
be verified?
Exercise 2
Check that equation (1) is correct for each of the cases listed in Exercise 1.
Exercise 3
Do the same with equation (2) by checking it case-by-case.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
Exercise 4
Do the same with equation (3) by checking it case-by-case.
Exercise 5
Write the expanded form of 8,374 using exponential notation.
Exercise 6
Write the expanded form of 6,985,062 using exponential notation.
Problem Set
Let 𝑥, 𝑦 be numbers (𝑥, 𝑦 ≠ 0). Simplify each of the following expressions of numbers.
1.
2.
12
𝑦
=
𝑦12
915 ∙
3.
1
=
915
4.
(7(123456.789)4 )0
=
22 ∙
1 5 1
∙2 ∙ 2 =
25
2
5.
𝑥 41 𝑦15
∙
=
𝑦15 𝑥 41
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
21
Name ____________________________ Period ____ Math Number ____
22
Lesson 5: Negative Exponents and the Laws of Exponents
If you raise a number to a negative exponent, then what do you hypothesize its value will be?
Question
3−2 =?
Hypothesis
Do others in class agree with you? YES _____ NO _____ SOME _____
What, if any, are some of the other hypotheses? ____________________
Let's investigate...
In math we may often use patterns to help us find a solution. So, ...
Power
34
33
35
Value
Meaning 3x3x3x3x3 3x3x3x3 3x3x3
32
3x3
31
3
30
3−1
3−2
3−3
1
What is happening to the value of the number as you move to the right in the table...
When you move _______ box to the right, the value of the number is _______ by 3.
What is happening to the value of the number as you move to the left in the table...
When you move _______ box to the left, the value of the number is _______ by 3.
Power
Value
25
24
23
22
21
20
2−1
2−2
2−3
Power
Value
15
14
13
12
11
10
1−1
1−2
1−3
Are numbers raised to a negative power less than zero? Yes ___ No ___
If not, then how would you describe them using math terms?
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
23
1
Definition: For any positive number 𝑥 and for any positive integer 𝑛, we define 𝑥 −𝑛 = 𝑥 𝑛.
Note that this definition of negative exponents says 𝑥 −1 is just the reciprocal
1
𝑥
of 𝑥.
As a consequence of the definition, for a positive 𝑥 and all integers 𝑏, we get
1
𝑥 −𝑏 = 𝑏
𝑥
1
Verify the general statement 𝑥 −𝑏 = 𝑥 𝑏 for 𝑥 = 3 and 𝑏 = 5.
Simplify as much as possible.
5−3 =
Let 𝑥 be a nonzero number.
𝑥 −3 =
1
=
89
Let 𝑥 be a nonzero number.
3 ∙ 2−4 =
Let 𝑥, 𝑦 be two nonzero numbers.
1
=
𝑥9
𝑥𝑦 −4 =
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
24
Practice on lessons 1-6: Match each number to a letter and write you answers on the
lines.
1] 8
2] 64
1
1 2
𝐴] (− )
2
B] −43
3] 4
C] (−5)2
4] -256
D] 102
5] 27
E] 23
6] 100
F] (−2)6
7] -64
8] 25
1
1 3
G] (2)
H] 32
9] 8
I] −28
10] 9
J] −(−3)3
1] _____
2] _____
3] _____
4] _____
5] _____
6] _____
7] _____
8] _____
9] _____
10] _____
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
Classwork
Definition: For any positive number 𝑥 and for any positive integer 𝑛, we define 𝑥−𝑛 =
Note that this definition of negative exponents says 𝑥−1 is just the reciprocal,
1
𝑥
1
𝑥𝑛
.
, of 𝑥.
As a consequence of the definition, for a positive 𝑥 and all integers 𝑏, we get
𝑥−𝑏 =
1
𝑥𝑏
.
Exercise 1
Verify the general statement 𝑥 −𝑏 =
1
𝑥𝑏
for 𝑥 = 3 and 𝑏 = −5.
Exercise 2
What is the value of (3 × 10−2 )?
Exercise 3
What is the value of (3 × 10−5 )?
Exercise 4
Write the complete expanded form of the decimal 4.728 in exponential notation.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
25
Name ____________________________ Period ____ Math Number ____
26
We accept that for positive numbers 𝑥, 𝑦 and all integers 𝑎 and 𝑏,
𝑥 𝑎 ∙ 𝑥 𝑏 = 𝑥 𝑎+𝑏
𝑎
(𝑥 𝑏 ) = 𝑥 𝑎𝑏
(𝑥𝑦)𝑎 = 𝑥 𝑎 𝑦 𝑎 .
We claim
𝑥𝑎
𝑥𝑏
𝑥 𝑎
= 𝑥 𝑎−𝑏
𝑥𝑎
(𝑦) = 𝑦𝑎
for all integers 𝑎, 𝑏.
for any integer 𝑎.
For Exercises 5–10, write an equivalent expression, in exponential notation, to the one given and simplify
as much as possible.
Exercise 5
Exercise 6
5−3 =
1
=
89
Exercise 7
Exercise 8
3 ∙ 2−4 =
Let 𝑥 be a nonzero number.
𝑥 −3 =
Exercise 9
Exercise 10
Let 𝑥 be a nonzero number.
Let 𝑥, 𝑦 be two nonzero numbers.
1
=
𝑥9
𝑥𝑦 −4 =
Exercise 11
Exercise 12
192
=
195
𝟏𝟕𝟏𝟔
=
𝟏𝟕−𝟑
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
27
Exercise 13
If we let 𝑏 = −1 in (11), 𝑎 be any integer, and 𝑦 be any positive number, what do we get?
Exercise 14
Show directly that
7 −4
(5)
=
7−4
5−4
.
Problem Set
1.
Compute: 33 × 32 × 31 × 30 × 3−1 × 3−2 =
Compute: 52 × 510 × 58 × 50 × 5−10 × 5−8 =
Compute for a nonzero number, 𝑎: 𝑎𝑚 × 𝑎𝑛 × 𝑎𝑙 × 𝑎 −𝑛 × 𝑎 −𝑚 × 𝑎−𝑙 × 𝑎0 =
2.
Without using (10), show directly that (17.6−1 )8 = 17.6−8 .
3.
Without using (10), show (prove) that for any whole number 𝑛 and any positive number 𝑦, (𝑦 −1 )𝑛 =
𝑦 −𝑛 .
4.
Show directly without using (13) that
2.8−5
2.87
= 2.8−12 .
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
28
Lesson 6: Proofs of Laws of Exponents
Class Discussion:
Classwork
The Laws of Exponents
For 𝑥, 𝑦 > 0, and all integers 𝑎, 𝑏, the following holds:
𝑥 𝑎 ∙ 𝑥 𝑏 = 𝑥 𝑎+𝑏
𝑎
(𝑥 𝑏 ) = 𝑥 𝑎𝑏
(𝑥𝑦)𝑎 = 𝑥 𝑎 𝑦 𝑎
Facts we will use to prove (11):
(A) (11) is already known to be true when the integers 𝑎 and 𝑏 satisfy 𝑎 ≥ 0, 𝑏 ≥ 0.
(B) 𝑥 −𝑚 =
(C)
1 𝑚
1
𝑥𝑚
for any whole number 𝑚.
1
(𝑥) = 𝑥 𝑚 for any whole number 𝑚.
Exercise 1
Show that (C) is implied by equation (5) of Lesson 4 when 𝑚 > 0, and explain why (C) continues to hold
even when 𝑚 = 0.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
29
Exercise 2
Show that (B) is in fact a special case of (11) by rewriting it as (𝑥 𝑚 )−1 = 𝑥 (−1)𝑚 for any whole number 𝑚,
so that if 𝑏 = 𝑚 (where 𝑚 is a whole number) and 𝑎 = −1, (11) becomes (B).
Exercise 3
Show that (C) is a special case of (11) by rewriting (C) as (𝑥 −1 )𝑚 = 𝑥 𝑚(−1) for any whole number 𝑚.
Thus, (C) is the special case of (11) when 𝑏 = −1 and 𝑎 = 𝑚, where 𝑚 is a whole number.
Exercise 4
Proof of Case (iii): Show that when 𝑎 < 0 and 𝑏 ≥ 0, (𝑥 𝑏 )𝑎 = 𝑥 𝑎𝑏 is still valid. Let 𝑎 = −𝑐 for some
positive integer 𝑐. Show that the left side and right sides of (𝑥 𝑏 )𝑎 = 𝑥 𝑎𝑏 are equal.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
30
Problem Set
5.
You sent a photo of you and your family on vacation to seven Facebook friends. If each of them
sends it to five of their friends, and each of those friends sends it to five of their friends, and those
friends send it to five more, how many people (not counting yourself) will see your photo? No friend
received the photo twice. Express your answer in exponential notation.
# of New People to View Your Photo
Total # of People to View Your Photo
6.
Show directly, without using (11), that (1.27−36 )85 = 1.27−36∙85 .
7.
Show directly that ( )
8.
Prove for any positive number 𝑥, 𝑥 −127 ∙ 𝑥 −56 = 𝑥 −183 .
9.
Prove for any positive number 𝑥, 𝑥 −𝑚 ∙ 𝑥 −𝑛 = 𝑥 −𝑚−𝑛 for positive integers 𝑚 and 𝑛.
2 −127
13
2 −56
∙ (13)
2 −183
= (13)
.
10. Which of the preceding four problems did you find easiest to do? Explain.
11. Use the properties of exponents to write an equivalent expression that is a product of distinct primes,
each raised to an integer power.
105 ∙ 92
=
64
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
31
MIXED PRACTICE
Please be very careful to show all work (SAW) in the space provided under each question.
Circle anything that you do not understand so that you remember to ask me.
Expression
Simplified Exponential Form
Simplified
32  34
24  28
24  24  04
3( x 2  1) when x = 2
4 x9
when x = 2
8 x12
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
Write each as a product of factors (expanded form).
1] 3x2 = _______________
2] (3x) 2 = _______________
3] -3x2 = _______________
4] (-3x) 2 = _______________
5] -(3x) 2 = _______________
6] -(-3x) 2 = _______________
Find each product or quotient and express using exponents.
7] 9 3  9 2  _______________
8] 7 3  7 = _______________
9] (n 4 )(n 4 ) = _______________
10]  3x 2 (4 x 3 ) = _______________
11]
38
 _______________
35
x3
12]
 _______________
x
( x) 5
13]
 _______________
(  x) 4
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
32
Name ____________________________ Period ____ Math Number ____
n 3 (n 5 )
14]
 _______________
n2
8n 2
15]
 _______________
32n
Write each expression using a positive exponent.
16] 5-2 = _______________
17] m-6 = _______________
18] (-7)-1 = _______________
19] (-3)-3 = _______________
20] (xy)-6 = _______________
Express in simplest terms.
n5
21] 5  _______________
n
22] 230 = _______________
23] c 3  c 0  _______________
24] 52  5 2  _______________
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
33
Name ____________________________ Period ____ Math Number ____
34
25] Simplify the following expression by two different methods:
(22 )3
Method 1:
Method 2:
If the following expressions were money, then which would you rather have AND why? Let
x=2
( x  22 )3 or x2  23
Make up a mathematical expression that is equal to 20 and uses multiplication, division, and
an exponent.
Which is greater, “two to the fourth” or “three to the third”? Explain.
3w  81, when w  _____
Two names for 63 are ______________ and ____________________.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
How much of an increase do you get if your allowance changes from $ 23 to $ 2 4 each
week?
Simplify:
7 20  730 (Leave in exponential form)
Simplify:
214  35  23  31  0 (Do NOT leave in exponential form)
Explain the shortcut for multiplying exponents with the same base.
Simplify:
2n  2 g  2m
Simplify. Leave in exponential form.
32  34 = __________
55  51
= __________
24  28 = __________
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
35
Name ____________________________ Period ____ Math Number ____
36
Simplify. Do NOT leave in exponential form.
62  33
= __________
24  24  04 = __________
83  53 = __________
Find the value of
way.
Find the value of
Find the value of
organized way.
3( x 2  1)
when x = 3. SHOW ALL WORK in a neat and organized
when x = 2. SHOW ALL WORK in a neat and organized way.
x3  x 2  x  10 when x = 3. SHOW ALL WORK in a neat and
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents
Name ____________________________ Period ____ Math Number ____
Find the value of
2n2  3n3
12n2
3n3
Find the value of
Find the value of
Find the value of
4n 6
20n 2
37
when n  2
when n  2
when n  2
4  ( x  2)4  9 when x  5
( x3  y 2  x)
Find the value of
when x = 3 and y = 5. SHOW ALL WORK in a neat
24  5
and organized way.
Grade 8, Module 1, Topic A, Lessons 1-6
MODULE 1: INTEGER EXPONENTS and SCIENTIFIC NOTATION
Topic A: Exponential Notation and Properties of Integer Exponents