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7ACC
Unit 4
Exponents
M
T
W
Date
5/5
5/6
5/7
T
F
5/8
5/9
M
T
W
T
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5/12
5/13
5/14
5/15
5/16
Lesson
1
2
3
4
Topic
Exponential Notation
Multiplying Exponents
Multiplying Exponents with Coefficients
Raising a Power to a Power
Quiz
Dividing Exponents
Zero and Negative
5
Zero and Negative
Application
Review
Test
Name: _____________________________
Teacher: _____________________________
Period: ____________
1
Lesson 1
Exponential Notation
Zero and Negative Powers
Vocabulary:
Base – When a number is raised to a power, the number that is used as a factor is the base.
Exponential Form – A number written with a base and an exponent.
Expanded form – A number written as the sum of the values of its digits.
Compute – Solve. Get an answer.
Vocabulary: Review Questions
26
4x² + 7
1) Name the variable _______
2) Name the coefficient _______
3) Name the exponent ________
4) Name the base ______
5) Name the constant ______
1) What is the base _________
2) What is the exponent ________
Part I: Exponential Notation
Examples: Write the following in exponential form:
𝟗
𝟗
𝟗
𝟗
1) 5 × 5 × 5 × 5 × 5 × 5 ____________
2)
× 𝟕 × 𝟕 × 𝟕 ____________
𝟕
3) 2 ∙ 2 ∙ 2 ∙ 2 ∙ 9 ∙ 9 ____________
4) 4 ∙ 4 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ____________
Write in expanded form:
5) 6 3 _________________________________
7) (−
4 5
)
11
= _________________________
6) (−2)6_________________________________
8) What do you think the value of n can be in 𝑥 𝑛
Will these products be positive or negative? How do you know?
9) (−𝟏)
× (−𝟏) × ⋯ × (−𝟏) = (−𝟏)𝟏𝟐
⏟
𝟏𝟐 𝒕𝒊𝒎𝒆𝒔
10) (−𝟏)
× (−𝟏) × ⋯ × (−𝟏) = (−𝟏)𝟏𝟑
⏟
𝟏𝟑 𝒕𝒊𝒎𝒆𝒔
2
Understanding Exponents:
Find the value of n:
11) 2𝑛 = 16
12) 3𝑛 = 27
13) Rewrite 8 in exponential notation using 2 as the base.
14) Rewrite 81 in exponential notation using 3 as the base.
Compute the value:
4
15) 2
= ________
1 3
17) 62 − 25 + 53 = ________
16) − ( ) = ________
2
Try These:
Write the following in exponential form:
1) 5 ∙ 𝑥 ∙ 𝑥 ∙ 5 ∙ 𝑥 ∙ 𝑥
2)
1
3
∙
1
3
∙
1
3
∙
1
3
∙
Write in expanded form and compute the value:
1 4
4) (2)
______________________ = _________
1
3
3) 𝑦 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑥
Find the value of n:
5) 2𝑛 = 256
6) Rewrite 125 in exponential notation using 5 as the base.
Find the value of n:
7)
2𝑛 =
32
Compute:
8) 23 + 32 = ________
9) 42 − 24 + 51 = ________
3
Exponent Notation: Check for Understanding
1)
4 × ⋯ × 4 = ______
⏟
3.6 × ⋯ × 3.6 = 3.647
⏟
2)
7 𝑡𝑖𝑚𝑒𝑠
_______ 𝑡𝑖𝑚𝑒𝑠
(−11.63) × ⋯ × (−11.63) =
3) ⏟
_________
12 × ⋯ × 12 = 1215
⏟
4)
_______𝑡𝑖𝑚𝑒𝑠
34 𝑡𝑖𝑚𝑒𝑠
(−5) × ⋯ × (−5) = ________
5) ⏟
10 𝑡𝑖𝑚𝑒𝑠
7) (−13)
× ⋯ × (−13) = ________
⏟
6 𝑡𝑖𝑚𝑒𝑠
9) ⏟
𝑥 ∙ 𝑥 ⋯ 𝑥 = _______
185 𝑡𝑖𝑚𝑒𝑠
7
7
6) 2⏟× ⋯ × 2 = ______
21 𝑡𝑖𝑚𝑒𝑠
1
1
8) (−
⏟ 14) × ⋯ × (− 14) = __________
10 𝑡𝑖𝑚𝑒𝑠
10)
𝑥 ∙ 𝑥 ⋯ 𝑥 = 𝑥𝑛
⏟
_______𝑡𝑖𝑚𝑒𝑠
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
11) ⏟
95 𝑡𝑖𝑚𝑒𝑠
12) (−1.8)
× (−1.8) × ⋯ × (−1.8) = (−1.8)122
⏟
122 𝑡𝑖𝑚𝑒𝑠
4
Lesson 1: Homework
Write the following in exponential form:
1) 3  3  3
2) 5 5  x  x  y  y  y
3)
3
7
∙
3
7
∙
3
7
∙
3
7
Compute:
4) 13 + 43
5) 22 − 41 + 23
6) A square has a length of 6.2 square feet.
If the area of a square formula is 𝐴 = 𝑠 2 ,
what is its area?
7) Find the value of x:
𝟎. 𝟒𝒙 − 𝟐(𝟎. 𝟓𝒙 + 𝟗) = −𝟑
8) On our way to the “Polynomial Mall” we decide to pick up a few items. The only way to purchase the items
is to simplify the cost of all the items you want. First a new skirt is perfect, the cost is 7𝑥 2 − 3. Next you see a
great pair of boots which cost 𝑥 2 + 9𝑥 + 5. Around the corner you notice a nice belt for your dad. It is a
bargain at 6𝑥 − 1. After a long day a shopping, you decide to pick up a snack. The cost of the snack is 𝑥 2 − 𝑥.
What is the grand total (simplified) of your polynomial shopping experience?
Fill in the blanks about whether the number is positive or negative.
9) If 𝑛 is a positive even number, then (−55)𝑛 is __________________________.
10) If 𝑛 is a positive odd number, then (−72.4)𝑛 is __________________________.
5
(−15) × ⋯ × (−15) = −156 . Is she correct? How do you know?
11) Josie says that ⏟
6 𝑡𝑖𝑚𝑒𝑠
12) Write an exponential expression with (−1) as its base that will produce a positive product.
13) Write an exponential expression with (−1) as its base that will produce a negative product.
14) Rewrite each number in exponential notation using 2 as the base. (Example: 2 = 21 )
a) 8 = ______
b) 16 = _____
c) 32 = _______
d) 64 = ______
e) 128 = _____
f) 256 = _______
15) Tim wrote 16 as (−2)4 . Is he correct?
16) Could −2 be used as a base to rewrite 32?
Why or why not?
6
Lesson 2
Laws of Exponents: Multiplying
Vocabulary:
Standard form - The way you write any number normally.
Rule: When multiplying exponents with the same base:
1 - Keep the Base
2 - Add the exponents
Examples: Simplify and write in standard form
1) 52 ∙ 53 = ___________
2) 73 ∙ 74 = ____________
3) 210 ∙ 216 _____________
Examples: Multiply using the laws of exponents.
4) 83 ∙ 85 = _______
5) 34 ∙ 34 = ______
6) 96 ∙ 9−3 = ______
7) (5) ∙ (5) = _______
8) 65 ∙ 6−5 = _____
9) 3−4 ∙ 3−5 ∙ 3 = ______
10) 2−2 ∙ 27 ∙ 20 = _____
11) 𝑎−1 ∙ 𝑎−3 ∙ 𝑎 = ______
12) 𝑥 2 ∙ 𝑥 5 = _____
2 3
2 7
Can the following expressions be simplified? If so, write an equivalent expression. If not, explain why not.
13) 72 ∙ 53 ∙ 7 = ______
15) 23 ∙ 42 = ______
* 14) 52 ∙ 125 = ______
16) 22 ∙ 53 ∙ 7 = ______
7
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
Try These: Multiply using the laws of exponents.
1) 29 ∙ 24 = _______
𝑥 4
𝑥 −1
5) (𝑦) ∙ (𝑦)
= _____
2) 4∙ 46 = _______
3) 5−5 ∙ 56 = ______
4) 92 ∙ 36 = ______
6) 7−8 ∙ 78 ∙ 7 = ______
7) 𝑦 4 ∙ 𝑦 9 = ______
8) 𝑥 3 ∙ 𝑥 7 =______
9) Which is 6 3 x 6 4 in standard form?
A. 3612
C. 612
10) Which is equal to 92?
A. 29
C. 34
B. 76
D. 67
B. 27
D. 92
Lesson 2: Classwork: Multiply using the laws of exponents. Rewrite as a positive exponent if necessary
1) 1013 ∙ 10−8 = _______
5) 6−7 ∙ 62 ∙ 6−4 = ______
2) 2−2 ∙ 2 = _______
1 6
B. 7
D. 49
11) What is the value of 3 4  3 7 ?
A. 3−3
C. 9−3
1 2
6) (2) ∙ (2) = ______ 7) 𝑐 −3 ∙ 𝑐 9 = ______
9) Which is (-7)2 in standard form?
A. -7
C. -49
3) 1−3 ∙ 19 ∙ 14 = ______
B. 33
D. 311
4) 84 ∙ 8−4 = _____
8) 𝑥 2 ∙ 𝑥 4 ∙ 𝑥 =___
10) Which shows (-3)2 in standard form?
A. 9
C. -9
B. -6
D. 6
12) The result of 8−4 comes from which multiplication
A. 83 ∙ 85
C. 83 ∙ 8−7
B. 34 ∙ 34
D. 34 ∙ 3−8
8
Lesson 2: Homework
Multiply using the laws of exponents.
1) 22 ∙ 25
2) 38  311
3) 44 ∙ 4
4) 6−3 ∙ 62
5) 37 ∙ 3−4
6) (−7) ∙ (−7)9
7) 𝑦 −4 ∙ 𝑦 −7
8) 60 ∙ 64
10) 𝑥 4 ∙ 𝑥 −4
11) 3 ∙ 3−3 ∙ 3 ∙ 30
12) (3)
14) 7 ∙ 7
15) (−9)−5 ∙ (−9)6
16) 𝑥 2 ∙ 𝑦 6
1 −2
9) (2)
1 1
∙ (2)
13) 𝑥 7 ∙ 𝑥 4
17) Which is -23 in standard form?
A. 8
C. 6
2 9
∙ (3)
18) Which shows 2-2 x 26 in exponential form?
A. 24
C. 2-8
B. -8
D. -6
19) Amy wrote these expressions:
2 −5
63
35
Part A: Write these expressions in order from least to greatest.
B. 2-4
D. 2-12
102
________
________
__________
Part B: Explain how you know your answer is correct____________________________________________
__________________________________________________________________________________________
20) Write in simplest form:
4h – 7h + 9 – 2h + 6 + 3h – 1
21) Which of the following equations has
Infinitely Many Solutions?
(1) 6x - 9 + 4x = 13 - x
(2) x - 11 = -x + 2x - 1
(3) 9 - 3x = 3x - 6x + 10 - 1
(4) 3 = 7x – x + 21
9
Check for Understanding:
Exercise 1
𝟏𝟒𝟐𝟑 × 𝟏𝟒𝟖 =
Exercise 5
Let 𝑎 be a number.
𝑎23 ∙ 𝑎8 =
Exercise 2
(−72)10 × (−72)13 =
Exercise 6
Let f be a number.
𝑓 10 ∙ 𝑓 13 =
Exercise 3
𝟓𝟗𝟒 × 𝟓𝟕𝟖 =
Exercise 7
Let 𝑏 be a number.
𝒃𝟗𝟒 ∙ 𝒃𝟕𝟖 =
Exercise 4
(−𝟑)𝟗 × (−𝟑)𝟓 =
Exercise 8
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
94 × 96 × 913 =
Exercise 10
23 × 25 × 27 × 29 =
Can the following expressions be simplified? If so, write an equivalent expression. If not, explain why not.
Exercise 11
Exercise 14
5
9
3
14
24 × 82 = 24 × 26 =
6 ×4 ×4 ×6 =
Exercise 12
(−4)2 ∙ 175 ∙ (−4)3 ∙ 177 =
Exercise 15
37 × 9 =
Exercise 13
152 ∙ 72 ∙ 15 ∙ 74 =
Exercise 16
54 × 211 =
10
Lesson 3
Multiplying Exponents with Coefficients
Raising a Power to a Power
Part 1: Multiplying Exponents with Coefficients
Rule: If there is a coefficient and exponents:
1 – Multiply Coefficients
2- Add exponents of like bases
Examples:
1) (5x²)(3x³)
2) (-6ab3)(-2a2b7)
3) (3ab)(-5a²bc³)
4) (2x-6y5)(-5x2y-3)
5) 7𝑥 2 ∙ 3𝑦 6
6) 5𝑐 −3 ∙ 3𝑐 9
7) 4𝑥 2 ∙ 7𝑥 4 ∙ 𝑥
8) 2𝑥 2 ∙ 5𝑥 7
9) 7𝑥 −2 ∙ −3𝑦 3
Part 2: Raising a Power to a Power
Vocabulary:
Power - _________________________________________________________________________________
Examples:
Simplify:
1) (52 )3 = ___________
2) (73 )4 = ____________
3) (210 )2 = _____________
11
Rule:
When raising a power to a power: Multiply the exponents
Examples:
4) (98 )4 = ______
5) (34 )2 = ______
6) (96 )2 = ______
7) (109 )4 = ______
8) (35 )3 = ______
9) (6−2 )3 = ______
10) (62 )2 ∙ 6−5 = ______
11) (27 )2 ∙ (2)−1 = _____
Rule: When raising a monomial to a power:
1 - Distribute the exponent to all terms.
2 - Use the rules of raising a power to a power to simplify.
12) (−4𝑥 8 )3
13) (3𝑦 7 )4
16) (−1𝑚4 )4
17) (3 𝑦 −5 )
1
2
14) (2𝑎11 )3
15) (5𝑥 3 )3
18) (2𝑥 −1 )3
19) (−2𝑏 −4 )4
Part 3: Distributive Property
20) 3( x + 8)
21) -2(𝑥 3 + 4)
22) - ( 3x – 5)
23) 3x(4𝑥 2 + 2x)
24) (x + 2)(x + 4)
25) (x+ 3)2
12
Try These:
Multiply using the laws of exponents.
1) (29 )5
2) (42 )3
3) (5−5 )3
4) (33 )3
5) (70 )7
6) (2𝑥 2 )3
7) (−3𝑦 5 )2
8) (𝑥 4 ∙ 𝑥 2 )2
9) (−2𝑥 −2 )3
10) (6−2 )3 ∙ (6−5 )
11) 6x3(5x2 – 3x)
12) (x + 2)2
3) (1−3 )4
4) (8−3 )4
Lesson 3: Classwork
Multiply using the laws of exponents.
1) (10−3 )2
2) (3−2 )4
5) (53 ∙ 5)2 ∙ 5−7
6) 9−2 ∙ 96
7) −(−5𝑎7 )2
8) (6𝑦 3 )3
9) (2𝑥 −1 ∙ 𝑥 3 )3
10) (−𝑏 −4 )4
11) Which is (2x2)1 in standard form?
12) Which shows (32)2 in standard form?
A. 2x3
A. 12
C. 2x2
B. 4x2
D. 4x3
13) Simplify: x(x + y)
A. x2 + y
C. 2x2y
B. x2 + xy
D. 2x + xy
C.
B. 81
1
D.
81
1
12
14) Simplify: (x + y)2
A. x2 + y2
B. x2 + 2xy + y2
13
Lesson 3: Homework
Multiply using the laws of exponents.
1) (84 )3
2) (2−1 )−3
5) (8−1 )−2
6) (37 )3 ∙ (3−2 )−4
9) (124 )4
10) 9 2  9 6
3) (3−3 )4
4) (7−2 )2
7) (91 )6
8)[(2) ]
11) (𝑦 15 )2
2
1 3
12) (3𝑥 5 )4
13) (3𝑦 3 ∙ −2𝑦)3
14) (5𝑥 −1 )2
15) Which is (7-2 )-1 in standard form?
16) Which shows (3x -2)2 in standard form?
A. -49
C. -14
B. 14
D. 49
A. -9x4
C. 3x4
B. 3x-4
D. 9x-4
17) Frank wrote the expression 9-2.
Part A: What is the value (compute) of the expression?_______________________________________
Part B: Is the expression (-9)2 equivalent to 9-2? Explain how you know._________________________
____________________________________________________________________________________
18) The formula for the volume of a rectangular
prism is V = LWH. If the L = 84 and W = 8−2
and the H = 80 . What is the volume in
exponential form?
19) In exponential form, what is the area of a
square that has length of 4−3 ?
14
Exercise 20
Let 𝑥 be a number. Simplify the expression of the following number:
(2𝑥 3 )(17𝑥 7 ) =
Exercise 21
Let 𝑎 and 𝑏 be numbers. Use the distributive law to simplify the expression of the following number:
𝑎(𝑎 + 𝑏) =
Exercise 22
Let 𝑎 and 𝑏 be numbers. Use the distributive law to simplify the expression of the following number:
𝑏(𝑎 + 𝑏) =
Exercise 23
Let 𝑎 and 𝑏 be numbers. Use the distributive law to simplify the expression of the following number:
(𝑎 + 𝑏)(𝑎 + 𝑏) =
Lesson 3: Check for Understanding
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)
=⏟
=𝑥
𝑚 𝑡𝑖𝑚𝑒𝑠
𝑚𝑛
(𝑛 𝑡𝑖𝑚𝑒𝑠)
𝑚 𝑡𝑖𝑚𝑒𝑠
Exercise 1
(153 )9 =
Exercise 3
(3.417 )4 =
Exercise 2
((−2)5 )8 =
Exercise 4
Let 𝑠 be a number.
(𝑠17 )4 =
15
Exercise 5
Sarah wrote that (35 )7 = 312 . Correct her mistake. Write an exponential expression using a base of 3 and
exponents of 5, 7, that would make her answer correct.
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
= 𝑥𝑛𝑦𝑛
Exercise 6
(11 × 4)9 =
Exercise 7
(32 × 74 )5 =
Exercise 8
Let 𝑎, 𝑏, and 𝑐 be numbers.
(32 𝑎4 )5 =
Exercise 9
Let 𝑥 be a number.
(5𝑥)7 =
Exercise 10
Let 𝑥 and 𝑦 be numbers.
(5𝑥𝑦 2 )7 =
Exercise 11
Let 𝑎, 𝑏, and 𝑐 be numbers.
(𝑎2 𝑏𝑐 3 )4 =
Exercise 12
𝑥 𝑛
Let 𝑥 and 𝑦 be numbers, 𝑦 ≠ 0, and let 𝑛 be a positive integer. How is (𝑦) related to 𝑥 𝑛 and 𝑦 𝑛 ?
Excerise 13
Show (prove) in detail why (𝑥𝑦𝑧)4 = 𝑥 4 𝑦 4 𝑧 4 for any numbers 𝑥, 𝑦, 𝑧
16
Lesson 4
Laws of Exponents: Dividing
Vocabulary:
Quotient – The answer to a division problem.
Rule: When dividing exponents with the same base:
1- Keep the base
2- Subtract the exponents
Examples:
Divide using the laws of exponents. Rewrite as a positive exponent if necessary.
58
1)
5
4)
3
57
53
_______________
= ______
𝑥10
7) 𝑥
10)
63
2)
_______________
62
= ______
𝑥3𝑦9
𝑦9
5)
8)
= ______
95
92
417
4 16
= ______
6)
= ______
9)
46 ∙52 ∙ (−1)7
11)
3) 2 5  2 3  ___________
4 5 ∙ (−1)4
73
79
= ______
𝑦 10
𝑦5
= ______
= ______
Rule: If there is a coefficient and exponents:
1 - Divide the coefficients
2 - Subtract exponents of like bases
12)
16)
5𝑥 3
5𝑥 2
9𝑎6
3𝑎2
= ______
= ______
13)
17)
12𝑥 5
−6𝑥 2
= ______
10𝑥 5 𝑦 12
20𝑥𝑦 8
= ______
14)
14𝑥 11
21𝑥 2
18)
4𝑥 50
2𝑦 25
= ______
15)
−18𝑥9
2𝑥14
= ______
4
= ______
19)
5𝑥 𝑦
𝑥14 𝑦8
= ______
17
Try These:
Divide using the laws of exponents. Rewrite as a positive exponent if necessary.
1)
139
133
= ______
5) 𝑥 8 ÷ 𝑥 4 = ______
2)
6)
35
3−4
6−2
6−6
= ______
3)
= ______
7)
24
24
𝑦6
= ______
4𝑤 3
4) 𝑦11 = ______
= ______
−2𝑤
8)
𝑥 12 ∙𝑦 7 ∙ 𝑝4
𝑥 4 ∙𝑦 7 ∙𝑝
= ___
Lesson 4: Classwork
Divide using the laws of exponents. Rewrite as a positive exponent if necessary.
1) 34 ÷ 3 = ______
5)
9)
𝑧 23
𝑧 −7
= ______
−32𝑥 8
−16𝑥 3
= ______
2)
6)
86
84
𝑥3
𝑥3
10)
= ______
3)
= ______
7)
74 ∙75 ∙ 34
(73 )2 ∙3
412
47
= ______
4𝑥 6
16𝑦 7
1 5
2
1 3
( )
2
= ______
4)
11)
= ______
312
= ______
8) 30𝑥 7 ÷ 5𝑥 4 = ____
( )
= ______
311
12)
𝑥5𝑦6
𝑥 −5 𝑦
= ______
18
Lesson 4: Homework
Divide using the laws of exponents. Rewrite as a positive exponent if necessary.
1)
5)
9)
84
82
= ______
911
99
𝑥3
𝑥
13)
= ______
= ______
6𝑥 5
3𝑥
= ______
4
2)
6)
48
43
= ______
710
7−9
10)
14)
3)
= ______
𝑚−2
𝑚−3
7)
= ______
5𝑎2
10𝑎 −3
53
59
10−4
10−9
11)
= ______
= ______
15)
= ______
𝑥4𝑦7
𝑥2𝑦3
= ______
57 ∙611 ∙ 𝑥 4
53 ∙6−7 ∙𝑥
= ______
4)
122
122
46
8)
4 13
12)
16)
= ______
= ______
𝑤 −4 𝑦 12
𝑥 −7 𝑦 3
(44 ∙4)2 ∙ 32
(4 2 )3 ∙3
= _____
= _____
Review Work:
Multiply using the laws of exponents. Rewrite as a positive exponent if necessary.
2
1
17) 10  10 = ______
3
6
18) x  x = ______
19) 26  2 2 =______
20) (32 ∙ 35 )2 = ______
2
3
21) What is the value of   in fraction form? _________________
5
22) Write in order from least to greatest? 42 , 40 , 4−1 , 41
_________________
23) Determine the missing (?) value in each:
A) (5? )3 = 512
B)
28
2?
= 29
C) (−2𝑚3 𝑛4 )? = −8𝑚9 𝑛12
19
Check for Understanding:
In general, if 𝑥 is nonzero and 𝑚, 𝑛 are positive integers, then
𝑥𝑚
= 𝑥 𝑚−𝑛
𝑥𝑛
𝑖𝑓 𝑚 > 𝑛
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
20
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
Exercise 28
Exercise 30
323 323
= 3 =
27
3
(−2)7 ∙ 955
=
(−2)5 ∙ 954
Exercise 31
Let 𝑥 be a number. Simplify the expression of each of the following numbers:
1.
2.
3.
5
𝑥3
5
𝑥3
5
𝑥3
(3𝑥 8 ) =
(−4𝑥 6 ) =
(11𝑥 4 ) =
21
Lesson 5
Zero and Negative Powers
Review:
Let us summarize our main conclusions about exponents. For any numbers 𝑥, 𝑦 and any positive integers 𝑚, 𝑛,
the following holds:
1)
𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛
Rule: ___________________________________________________
2)
(𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛
Rule: ___________________________________________________
3)
(𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛
Rule: ___________________________________________________
And if we assume 𝑥 > 0 in equation (4) and 𝑦 > 0 in equation (5 and 7) below, then we also have:
4)
𝑥𝑚
𝑥𝑛
= 𝑥 𝑚−𝑛 , 𝑚 > 𝑛
𝑥 𝑛
𝑥𝑛
Rule: ___________________________________________________
5) (𝑦) = 𝑦 𝑛
Rule: ___________________________________________________
6) 𝑥 0 = 1
Rule: ___________________________________________________
7)
𝑥 −2
𝑦 −3
=
𝑦3
𝑥2
Rule: ___________________________________________________
Part 1: Zero Exponent
Review:
1)
24
22
2)
2
2
3)
24
24
Rule
Any Number to the Zero Power equals __________
22
Examples:
4) 270
5) (3x)0
6) 3x0
7) (x4y)0
8) 8(x2y3)0
9) 4x(x9y5)0
Part 2: Negative Exponents
Review:
34
1) 2
3
34
2)
3)
34
34
36
Remember: All negative
exponents represent a fraction
or decimal.
𝟏
Definition: For any positive number 𝑥 and for any positive integer 𝑛, we define 𝒙−𝒏 = 𝒙𝒏 .
Note that this definition of negative exponents says 𝒙−𝟏 is just the reciprocal
𝟏
𝒙
of 𝒙.
𝟏
As a consequence of the definition, for a positive 𝑥 and all integers 𝑏, we get 𝒙−𝒃 = 𝒙𝒃
Write the following as a positive exponent:
Rule:
To convert the exponent from a negative to a positive, take the reciprocal of the
base and make the exponent positive.
Examples:
4) (4)−4
3 −1
________
5) ( )
7) 4−1 = ________
8) ( )
4
5 −2
6
________
= ________
6)
5−4
2−6
9) 10−3
________
= ________
23
Using Exponent Rules:
Remember: When you multiply powers with the same base, you add the exponents.
When you divide powers with the same base, you subtract the exponents.
Write each answer with a positive exponent
10) x5 · x9
11) 102 · 103
12) x6 · x -4
13) x -2 · x -3
14)
10 3  10 7
5
3
3
15)
2
16) ( x y )( x y )
18)
x8  x 2
20) x  x
8
2
x 4  x
17) (2 x y )(5x y )
4
19)
2
3
5
x4  x6
21) 10  10
2
5
24
Try These:
Simplify:
1) 50
5)
2) 4x0
3 x( y ) 0
6) (3xy)
3) (4x)0
4) 3(ab)0
9) 5−2
10) (2)
0
Write the following as a positive exponent:
1 −3
8) 9−2
7) (2)
1 −4
Write each answer with a positive exponent
11)
x 2  x5
12)
15) (4 x )(3x )
8
3
66  64
5
3
13)
3
x5  x5
20)
x 4  x 5
23)
(3x)0
24)
3x0
14)
17) x  x
4
9
16) (7 x y )( x y )
19)
x 6  x6
21)
32  32
6
x 7  x9
18) x  x
22)
4
x500  x500
25
Lesson 5: Homework
1)
30
2)
2x0
3)
(2 x) 0
4)
4( xy )0
5)
4a(b)0
6)
(4ab)0
7) x  x
3
10)
7
6
x 3  x 5
19) (8x)
22)
5
x10  x5
13) x  x
16)
8) 9  9
8
0
(2mn)0
11)
4
(6x9 )(8x2 )
14) x  x
17)
9) x  x
7
5
104  104
12)
4
(3x4 y 6 )( x3 y 4 )
15) x  x
7
7
18)
x40  x40
20)
8x0
21)
9x0
23)
0
9
24)
9
0
26
25) 5−3
28)
2 −4
26) 12−1
4−3
29)
7−2
27) (3)
1
8−2
Compute:
1 0
30) 93
31) 50
33) 22 + 23 + 91
34) 5−2 − 33
35)
Example
32) 32 + (9)
Write with a positive
exponent.
100
10−1
10−2
10−3
36) Which is 6 3 x 6 4 in standard form?
A. 3612
C. 612
B. 367
A. -729
D. 67
C.
38) Which is (-7)0 in standard form?
A. -7
C. 0
C.
1
9
1
27
A. 9
1
1
D.
7
D.
1
729
C.
9
B. 6
D.
1
6
41) The result of 8−4 comes from which multiplication
B. 27
D.
B. -27
39) Which shows (-3)2 in standard form?
B. 1
40) What is the value of 3 4  3 7 ?
A. -81
37) Which shows 9-3 in standard form?
1
27
A. 8−12 ∙ 8−8
B. 84 ∙ 8−4
C. 8−2 ∙ 8−2
D. 8−4 ∙ 8−8
27