exponents bundle 1 - patchogue-medford school district · 2019-09-12 · exponent review- remember...
TRANSCRIPT
CCSS.Math.Content.8.EE.A.1
Know and apply the properties of integer exponents
to generate equivalent numerical expressions.
(These activities include Positive Exponents only.)
By Barbara Robinson-Rockin’ Middle School Math
https://www.teacherspayteachers.com/Store/Rockin-Middle-School-Math
Exponents Bundle 1
Property Investigations, Notes, Practice, Games,
Homework, and Test
This Bundle includes the following activities:
1. Review of Exponents Practice Sheet
2. 3 Property Investigations (Discoveries):
Product of Powers Product of Quotients Power of Power
3. Interactive Notes: Includes a review of exponents and covers the 3 rules. Practice is provided on the notes
4. A Practice worksheet which covers Rule #1 and #2 only. You will probably
not cover all the rules on the first day. (This is two half sheets on one page) 5. A homework assignment which covers Rule #1 and #2 only. (This is two
half sheets on one page) 6. A Practice worksheet that covers all 3 rules 7. A Homework assignment that cover all 3 rules 8. A Partner Check practice 9. Game: I have, Who Has 10. Game: Connect 4 11. A Quiz or Test If you liked this Exponent Bundle, please see my Negative and Zero Exponent Bundle #2. You can find it at: https://www.teacherspayteachers.com/Product/Negative-and-Zero-Exponents-Bundle-2039460
Name ___________________________________ Date _________________________
Exponents Review
Write each number using exponents.
1) 6 x 6 = _________ 2) -5 x -5 x -5 x -5 = ____________
3) 27 = _________ 4) - (4 x 4 x 4) = ____________
5) a x a x a = _________ 6) 1
2 x
1
2 x
1
2 x
1
2 x
1
2 x
1
2 = ____________
7) -122 base _____ Expanded _______________ 8) (1
2)3 base ______ Expanded ________________
Exponent ______ Standard form ________ Exponent _______ Standard form __________
9) (-9)3 base _______ Expanded _______________________
Exponent _______ Standard form ___________
Write in expanded form.
10) 84 ______________________ 11) -32 ________________________________
12) 8 squared ____________________ 13) (-7)5 ________________________________
14) 7 cubed ______________________ 15) (3
4)5 ________________________________
Write in standard form.
16) 81 = ___________ 17) -24 = ______________
18) 4 squared = ___________ 19) (-2)4 = ______________
20) 5 cubed = ___________ 21) (-3)3 = ______________
22) (3
5)2 = ___________ 23) -33 = ______________
Write each expression in standard form. Then compare using <, >, or =.
24) 32 ______ 23 25) 73 ______ 7 x 7 x 7
26) 53 ______ 5 x 3 27) (-3)2 ______ -24
28) 2 squared ______ 2 cubed 29) 43 ______ 4 cubed
Circle the correct answer:
30) 48 in exponential notation: 86 68 480 481
31) Base is 4, Exponent is 5: 45 4 x 5 45 54
32) 216 in exponential notation: 63 36 2160 216
33) 23 + 34 57 2334 89 18
Key: Exponents Review
Write each number using exponents.
1) 6 x 6 = 62 2) -5 x -5 x -5 x -5 = (-5)4
3) 27 = 271 or 33 4) - (4 x 4 x 4) = -43
5) a x a x a = a3 6) 1
2 x
1
2 x
1
2 x
1
2 x
1
2 x
1
2 = (
𝟏
𝟐)6
7) -122 base 12 Expanded – (12 x 12) 8) (1
2)3 base (
𝟏
𝟐) Expanded (
𝟏
𝟐) 𝐱 (
𝟏
𝟐) 𝐱 (
𝟏
𝟐)
Exponent 2 Standard form -144 Exponent 𝟑 Standard form 𝟏
𝟖
9) (-9)3 base −𝟗 Expanded (−𝟗) 𝐱 (−𝟗) 𝐱 (−𝟗)
Exponent 3 Standard form -729
Write in expanded form.
10) 84 = 8 x 8 x 8 x 8 11) -32 = -(3 x 3)
12) 8 squared = 8 x 8 13) (-7)5 = (-7 x -7 x -7 x -7 x -7)
14) 7 cubed = 7 x 7 x 7 15) (3
4)5 = (
𝟑
𝟒) 𝐱 (
𝟑
𝟒) 𝐱 (
𝟑
𝟒) 𝐱 (
𝟑
𝟒) 𝐱 (
𝟑
𝟒)
Write in standard form.
16) 81 = 8 17) -24 = -16
18) 4 squared = 16 19) (-2)4 = 16
20) 5 cubed = 125 21) (-3)3 = -27
22) (3
5)2 =
𝟗
𝟐𝟓 23) -33 = -27
Write each expression in standard form. Then compare using <, >, or =.
24) 32 > 23 25) 73 = 7 x 7 x 7
26) 53 > 5 x 3 27) (-3)2 > -24
28) 2 squared < 2 cubed 29) 43 = 4 cubed
Circle the correct answer:
30) 48 in exponential notation: 86 68 480 481
31) Base is 4, Exponent is 5: 45 4 x 5 45 54
32) 216 in exponential notation: 63 36 2160 216
33) 23 + 34 57 2334 89 18
KEY
Investigating Exponent Properties: Product of Powers
Expression Expand the expression Simplified
Expression 𝒂𝒃
52 ∙ 54 𝟓 ∙ 𝟓 ∙ 5 ∙ 5 ∙ 5 ∙ 5 56
84 ∙ 85
(−4)3 ∙ (−4)5
74 ∙ 74 ∙ 7
𝑥3 ∙ 𝑥7
5𝑥4𝑦2 ∙ 3𝑥6𝑦9
Try this one without expanding it.
Do you see a pattern that relates the expression in the first column to the
equivalent simplified expression? Write in words what you notice about the
exponents.
Try these on your own
715
9𝑥2𝑦 ∙ 8𝑥0𝑦5
Here is the formal rule when multiplying exponential expressions with
the same base:
Investigating Exponent Properties: Quotient of Powers
Expression Expanded form Cancel factors Simplified
Expression 𝒂𝒃
43
42
4 ∙ 4 ∙ 4
4 ∙ 4
4∙4∙𝟒
4∙4 41
36
33
15
14
𝑥9
𝑥7
615𝑦45
67𝑦25 Try this one without expanding it.
Do you see a pattern that relates the expression in the first column to the
equivalent simplified expression? Write in words what you notice about the
exponents.
Try these on your own
Here is the formal rule when dividing exponential expressions with the
same base:
97
86𝑦5
83𝑦3
Investigating Exponent Properties: Power of a Power
Expression Expanded Form Expand further Simplified
Expression 𝒂𝒃
(92)3 92 ∙ 92 ∙ 92 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 96
(83)4
[(−6)2]5
(𝑥3)3
(𝑥4 ∙ 𝑦2)3
(𝑎20)32 Try this one without expanding it.
Do you see a pattern that relates the expression in the first column to the
equivalent simplified expression? Write in words what you notice about the
exponents.
Try these on your own
424
(𝑥2𝑦5𝑧3)3
Here is the formal rule when multiplying exponential expressions by a
power:
Key: Investigating Exponent Properties: Product of Powers Expression Expand the expression Simplified
Expression 𝒂𝒃
52 ∙ 54 𝟓 ∙ 𝟓 ∙ 5 ∙ 5 ∙ 5 ∙ 5 56
84 ∙ 85 𝟖 ∙ 𝟖 ∙ 𝟖 ∙ 𝟖 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 89
(−4)3 ∙ (−4)5 (−𝟒) ∙ (−𝟒) ∙ (−𝟒) ∙ (−4) ∙ (−4) ∙ (−4)∙ (−4) ∙ (−4)
(−4)8
74 ∙ 74 ∙ 7 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 𝟕 79
𝑥3 ∙ 𝑥2 𝒙 ∙ 𝒙 ∙ 𝒙 ∙ 𝑥 ∙ 𝑥 𝑥5
5𝑥4𝑦2 ∙ 3𝑥6𝑦9
Try this one without expanding it. 15𝑥10𝑦11
Do you see a pattern that relates the expression in the first column to the
equivalent simplified expression? Write in words what you notice about the
exponents.
When you multiply numbers that have the same base, add the exponents.
Try these on your own
Answers will vary
715
9𝑥2𝑦 ∙ 8𝑥0𝑦5 9 ∙ 8 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 72𝑥2𝑦6
Here is the formal rule when multiplying exponential expressions with
the same base: 𝒂𝒎 ∙ 𝒂𝒏 = 𝒂𝒎+𝒏
Key: Investigating Exponent Properties: Quotient of Powers
Expressio
n Expanded form Cancel factors
Simplified
Expression 𝒂𝒃
43
42
4 ∙ 4 ∙ 4
4 ∙ 4
4 ∙ 4 ∙ 𝟒
4 ∙ 4 41
36
33
3∙3∙3∙𝟑∙𝟑∙𝟑
3∙3∙3 33
15
14
1∙1∙1∙1∙𝟏
1∙1∙1∙1 11
𝑥9
𝑥7
𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝒙∙𝒙
𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥
𝑥2
615𝑦45
67𝑦25
Try this one without expanding it.
68𝑦20
Do you see a pattern that relates the expression in the first column to the
equivalent simplified expression? Write in words what you notice about the
exponents.
When you divide numbers that have the same base, subtract the
exponents. Try these on your own
Answers will vary
76
86𝑦5
83𝑦3 8∙8∙8∙8∙8∙8∙𝑦∙𝑦∙𝑦∙𝑦∙𝑦
8∙8∙8∙𝑦∙𝑦∙𝑦 83𝑦2
Here is the formal rule when multiplying exponential expressions with
the same base: 𝒂𝒎
𝒂𝒏 = 𝒂𝒎−𝒏
Key: Investigating Exponent Properties: Power of a Power
Expression Expanded Form Expand further Simplified
Expression 𝒂𝒃
(92)3 92 ∙ 92 ∙ 92 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 96
(83)4 83 ∙ 83 ∙ 83 ∙ 83 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8 ∙ 8
∙ 8
812
[(−6)2]5 (−6)2 ∙ (−6)2 ∙ (−6)2
∙ (−6)2 ∙ (−6)2
(−6) ∙ (−6) ∙ (−6) ∙ (−6)
∙ (−6) ∙ (−6) ∙ (−6) ∙ (−6)
∙ (−6) ∙ (−6)
(−6)10
(𝑥3)3 𝑥3 ∙ 𝑥3 ∙ 𝑥3 𝑋 ∙ 𝑋 ∙ 𝑋 ∙ 𝑋 ∙ 𝑋 ∙ 𝑋 ∙ 𝑋 ∙ 𝑋 ∙ 𝑋 𝑥9
(𝑥4 ∙ 𝑦2)3 (𝑥4 ∙ 𝑦2) ∙ (𝑥4 ∙ 𝑦2) ∙
(𝑥4 ∙ 𝑦2)
𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥
∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 𝑥12𝑦6
(𝑎20)32 Try this one without expanding it. 𝑎640
Do you see a pattern that relates the expression in the first column to the equivalent
simplified expression? Write in words what you notice about the exponents.
When you multiply a power of a number times a power, multiply the exponents.
Try these on your own
Answers will vary 424
(𝑥2𝑦5𝑧3)3 𝑥6𝑦15𝑧9
Here is the formal rule when multiplying exponential expressions by a
power: (𝒂𝒎)𝒏 = 𝒂𝒎 ∙ 𝒏
Notes: Exponential Rules
Exponent Review- Remember two things:
1. You never multiply a base by its exponent. The exponent tells you how many times to multiply
the base by itself. ∗ 43 is NOT 4 x 3. It is 4 x 4 x 4 = 64
2. If a base is negative, it must be in parentheses to use it when you multiply. Otherwise, your
answer will always be negative.
* (-3)4 means_________________________________________________________________
* −34 means __________________________________________________________________
YOUR TURN: Evaluate the exponential expression.
1. 53 = _________ 2. (2
3)
3 = _________ 3. (−6)3 = _________
4. 93 = _________ 5. −22 = _________ 6. 1.13 = _________
RULES OF EXPONENTS
Rule 1: Product of Powers 𝑎𝑚 ∙ 𝑎𝑛 = _____________
When ___________________ exponents with the same base, ________________ the exponents.
Ex: 43 ∙ 42 = ______ Why?
YOUR TURN: Simplify using exponents in your answer.
1) 32 ∙ 35 = _______ 2) (−4)3 ∙ (−4)8 = ________
3) (4)(46) = _________ 4) (−5)7 × (−5)3 × (−5)2 = ____________
Rule 2: Quotient of Powers 𝑎𝑚
𝑎𝑛= ___________
When ___________________ exponents with the same base, ______________ the exponents.
Ex: 36
34= _________ Why?
YOUR TURN: Simplify using exponents in your answer.
1) 69
65 = ________ 2)
(−18)10
(−18)2 = _______
3) 108 ÷ 104 = _________ 4) 95 ÷ 9 = _________
Rule 3: Power of a Power (𝑎𝑚)𝑛 = ____________
When finding the power of a power, ________________ the exponents.
Ex: = (63)4 = _______ Why?
YOUR TURN: Simplify using exponents in your answer.
1) (84)3 = __________ 2) (152)8 = __________
3) [(−4)6]2 = _________ 4) (93)5 = __________
REVIEW:
1) (−15)6 ∙ (−15)3 ∙ (−15) = _______ 2) [(−8)7]6 = _________
3) (−5)⁸
(−5)³ = _____ 4) (78)2 = ________
5) (41)5 = ________ 6) 34 ∙ 36 = __________
7) 65 × 62 = _________ 8) 1012
104 = __________
9) (−12)10 ÷ (−12)2 = __________ 10) 85 × 8 = ___________
Challenge:
11) 102 x 106
107 = __________ 12) 54 ∙ 56
54 = __________
13) 75x 73
7 x 72 = __________ 14) 47∙ 43
48÷ 42 = ___________
15) (36)2
(33)4 = __________ 16) 28∙ 23 x 2
26 ∙ 23 ∙ 22 = ___________
Key: Notes Exponential Rules
Exponent Review- Remember two things
1. You never multiply a base by its exponent. The exponent tells you how many times to multiply
the base by itself. ∗ 43 is NOT 4 x 3. It is 4 x 4 x 4 = 64
2. If a base is negative, it must be in parentheses to use it when you multiply. Otherwise, your
answer will always be negative.
* (-3)4 means -3 x -3 x -3 x -3 = 81
* −34 means negative or the opposite of (3 x 3 x 3 x 3) = -81
YOUR TURN: Evaluate the exponential expression.
1. 53 = 125 2. (2
3)
3 =
8
27 3. (−6)3 = -216
4. 93 = 729 5. −22 = -4 6. 1.13 = 1.331
RULES OF EXPONENTS
Rule 1: Product of Powers 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
When multiplying exponents with the same base, add the exponents.
Ex: 43 ∙ 42 = 45 Why? (4 x 4 x 4) x (4 x 4) = 45 or 43+2 = 45
YOUR TURN: Simplify using exponents in your answer.
1) 32 ∙ 35 = 37 2) (−4)3 ∙ (−4)8 = (−4)11
3) (4)(46) = 47 4) (−5)7 × (−5)3 × (−5)2 = (−5)12
Rule 2: Quotient of Powers 𝑎𝑚
𝑎𝑛= 𝑎𝑚−𝑛
When dividing exponents with the same base, subtract the exponents.
Ex: 36
34= 32 Why?
3 𝑥 3 𝑥 3 𝑥 3 𝑥 3 𝑥 3
3 𝑥 3 𝑥 3 𝑥 3 = 32 or 36−4 = 32
YOUR TURN: Simplify using exponents in your answer.
2) 69
65 = 64 2)
(−18)10
(−18)2 = (−18)8
3) 108 ÷ 104 = 104 4) 95 ÷ 9 = 94
KEY
Rule 3: Power of a Power (𝑎𝑚)𝑛 = 𝒂𝒎 𝐱 𝒏
When finding the power of a power, multiply the exponents.
Ex: (63)4 = 𝟔𝟏𝟐 Why? 𝟔𝟑 x 𝟔𝟑 x 𝟔𝟑 x 𝟔𝟑 = (6x6x6) x (6x6x6) x (6x6x6) x (6x6x6) = 𝟔𝟏𝟐
𝟔𝟑 𝐱 𝟒 = 𝟔𝟏𝟐
YOUR TURN: Simplify using exponents in your answer.
2) (84)3 = 812 2) (152)8 = 1516
4) [(−4)6]2 = (−4)12 4) (93)5 = 915
Your Turn Review:
1) (−15)6 ∙ (−15)3 ∙ (−15) = (−15)10 2) [(−8)7]6 = (−8)42
3) (−5)⁸
(−5)³ = (−5)5 4) (78)2 = 716
5) (41)5 = 45 6) 34 ∙ 36 = 310
7) 65 × 62 = 67 8) 1012
104 = 108
9) (−12)10 ÷ (−12)2 = (−12)8 10) 85 × 8 = 86
Challenge:
11) 102 x 106
107 = 101 12)
54 ∙ 56
54 = 56
13) 75x 73
7 x 72 = 75 14)
47∙ 43
48÷ 42 = 44
15) (36)2
(33)4 = 30 = 1 16) 28∙ 23 ∙ 2
26 ∙ 23 ∙ 22 = 21
KEY
Name _______________________________ Date ___________________
Practice: Exponents Review and Rules #1 and #2
Write each number using exponents.
1) 5 x 5 = _______ 2) -5 x -5 x -5 x -5 = _______
3) - (2 x 2 x 2) = _______ 4) n x n x n x n x n x n = _______
Evaluate:
5) -83 = _______ 6) (-3)2 = _______ 7) (-4)3 =________ 8) -162 = _______
9) (-7)3 = _______ 10) (-12)2 = _______ 11) (-10)5 = _______ 12) -14 = _______
Use the rules of exponents to simplify the expression with exponents:
13) 64 • 62 = _______ 14) 31 • 32 = _______ 15) 811 • 84 = _______ 16) 90 • 94 ______
17) 48 • 48 = _______ 18) 15 • 13 • 16 = _______ 19) 2
6
10
10 = _______ 20)
1
3
5
5 = _______
21) 4
5
8
8 =_______ 22)
2
4
2
2 = _______ 23)
2
7
7
7 = _______ 24)
4
10
4
4 = _______
Name _______________________________ Date ___________________
Practice: Exponents Review and Rules #1 and #2
Write each number using exponents.
1) 5 x 5 = _______ 2) -5 x -5 x -5 x -5 = _______
3) - (2 x 2 x 2) = _______ 4) n x n x n x n x n x n = _______
Evaluate:
5) -83 = _______ 6) (-3)2 = _______ 7) (-4)3 =________ 8) -162 = _______
9) (-7)3 = _______ 10) (-12)2 = _______ 11) (-10)5 = _______ 12) -14 = _______
Use the rules of exponents to simplify the expression with exponents:
13) 64 • 62 = _______ 14) 31 • 32 = _______ 15) 811 • 84 = _______ 16) 90 • 94 ______
17) 48 • 48 = _______ 18) 15 • 13 • 16 = _______ 19) 2
6
10
10 = _______ 20)
1
3
5
5 = _______
21) 4
5
8
8 =_______ 22)
2
4
2
2 = _______ 23)
2
7
7
7 = _______ 24)
4
10
4
4 = _______
Key to Practice: Exponents Review and Rules #1 and #2
Write each number using exponents.
1) 5 x 5 = 52 2) -5 x -5 x -5 x -5 = (-5)4
3) - (2 x 2 x 2) = -23 4) n x n x n x n x n x n = n6
Evaluate:
5) -83 = -512 6) (-3)2 = 9 7) (-4)3 = -64 8) -162 = -256
9) (-7)3 = -343 10) (-12)2 = 144 11) (-10)5 = -100,000 12) -14 = -1
Use the rules of exponents to simplify the expression with exponents:
13) 64 • 62 = 66 14) 31 • 32 = 33 15) 811 • 84 = 815 16) 90 • 94 = 94
17) 48 • 48 = 416 18) 15 • 13 • 16 = 114 19) 2
6
10
10 = 104 20)
1
3
5
5 = 52
21) 4
5
8
8 = 81 22)
2
4
2
2 = 22 23)
2
7
7
7 = 75 24)
4
10
4
4 = 46
KEY
Name: ________________________________ Date: ____________________
Homework: Exponents Review and Rules #1 and #2
Simplify:
1. (3
5)
2= _______ 2. -72 = _______ 3. 31 = _______ 4. (-7)2 = _______
Write the answer in Exponential form:
5. 68 ÷ 65 = _______ 6. 8
8
6
6 = _______ 7. (-8)4 • (-8)4 = _______ 8.
7
8
7
7= _______
9. 74 x 73 = _______ 10. (-7)4 ÷ (-7) 3 = _______ 11. 84 • 8 = ______ 12. 4
5
8
8 = _______
13. 4
8
3
3 = _______ 14. 62 x 69 x 6 = _______ 15. (-2)4 • (-2)4 = _______ 16.
(−8)5
(−8)2= _______
17. (-3)4 x (-3)4 x (-3)4 x (-3)4 = _____ 18. 5 cubed =_____ 19. 8
8
3
3=______ 20. 4 squared = _____
Name: ________________________________ Date: ____________________
Homework: Exponents Review and Rules #1 and #2
Simplify:
1. (3
5)
2= _______ 2. -72 = _______ 3. 31 = _______ 4. (-7)2 = _______
Write the answer in Exponential form:
5. 68 ÷ 65 = _______ 6. 8
8
6
6 = _______ 7. (-8)4 • (-8)4 = _______ 8.
7
8
7
7= _______
9. 74 x 73 = _______ 10. (-7)4 ÷ (-7) 3 = _______ 11. 84 • 8 = ______ 12. 4
5
8
8 = _______
13. 4
8
3
3 = _______ 14. 62 x 69 x 6 = _______ 15. (-2)4 • (-2)4 = _______ 16.
(−8)5
(−8)2= _______
17. (-3)4 x (-3)4 x (-3)4 x (-3)4 = _____ 18. 5 cubed =_____ 19. 8
8
3
3=______ 20. 4 squared = _____
Key to Homework: Exponents Review and Rules #1 and #2
Simplify:
1. (3
5)
2= 𝟗
𝟐𝟓 2. -72 = -49 3. 31 = 𝟑 4. (-7)2 = 49
Write the answer in Exponential form:
5. 68 ÷ 65 = 63 6. 8
8
6
6 = 60 7. (-8)4 • (-8)4 = (-8)8 8.
7
8
7
7= 71
9. 74 x 73 = 77 10. (-7)4 ÷ (-7) 3 = (-7)1 11. 84 • 8 = 85 12. 4
5
8
8 = 81
13. 4
8
3
3 = 34 14. 62 x 69 x 6 = 612 15. (-2)4 • (-2)4 = (-2)8 16.
(−8)5
(−8)2= (-8)3
17. (-3)4 x (-3)4 x (-3)4 x (-3)4 = (-3)16 18. 5 cubed =53 19. 8
8
3
3= 30 20. 4 squared = 42
KEY
Name _________________________________________ Date ___________________
Practice: Rules of Exponents
Review: Write the answer in standard form:
1) 53 = 2) 8 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 =
3) (−6)2 = 4) 9 𝑐𝑢𝑏𝑒𝑑 =
5) − 44 = 6) 21 =
7) − 43 = 8) (−3)3 =
Rule #1: Product of Powers Property 𝒂𝒎 ∙ 𝒂𝒏 = 𝒂𝒎+𝒏
9) 53 × 56 = 10) 72 ∙ 77 ∙ 73 =
11) 3 × 34 = 12) (−8)11 ∙ (−8)2 =
13) 9 × 92 = 14) 𝑥8 ∙ 𝑥8 =
Rule #2: Quotient of Powers Property 𝒂𝒎
𝒂𝒏= 𝒂𝒎−𝒏
15) 53
51= 16)
(−7)6
(−7)2=
17) 24
22= 18)
𝑛15
𝑛5=
19) 85 ÷ 84 = 20) (−9)5 ÷ (−9)5 =
Rule #3: Power of Power Property (𝒂𝒎)𝒏 = 𝒂𝒎∙𝒏
21) (53)6 = 22) (𝑛3)5 =
23) (22)6 = 24) [(−5)3]2 =
Mixed Practice/Review:
25) 32 ∙ 3 ∙ 35 = 26) 610
65=
27) 𝑛9 ÷ 𝑛2 = 28) 58 × 52 =
29) (23)3 = 30) [(−4)2]9 =
Make up two problems of your own examples for Rule #1. (answer them too).
31) 32)
Make up two problems of your own examples for Rule #2. (answer them too).
33) 34)
Make up two problems of your own examples for Rule #3. (answer them too).
35) 36)
Name _________________________________________ Date ___________________
Answer Key: Rules of Exponents
Review: Write the answer in standard form:
2) 53 = 125 2) 8 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 = 64
3) (−6)2 = 36 4) 9 𝑐𝑢𝑏𝑒𝑑 = 729
5) − 44 = -256 6) 21 = 2
7) − 43 = -64 8) (−3)3 = -27
Rule #1: Product of Powers Property 𝒂𝒎 ∙ 𝒂𝒏 = 𝒂𝒎+𝒏
9) 53 × 56 = 59 10) 72 ∙ 77 ∙ 73 = 712
11) 3 × 34 = 35 12) (−8)11 ∙ (−8)2 = (−8)13
13) 9 × 92 = 93 14) 𝑥8 ∙ 𝑥8 = 𝑥16
Rule #2: Quotient of Powers Property 𝒂𝒎
𝒂𝒏= 𝒂𝒎−𝒏
15) 53
51= 52 16)
(−7)6
(−7)2= (−7)4
17) 24
22= 22 18)
𝑛15
𝑛5= 𝑛10
19) 85 ÷ 84 = 81 20) (−9)5 ÷ (−9)5 = (−9)0 which = 1
Rule #3: Power of Power Property (𝒂𝒎)𝒏 = 𝒂𝒎∙𝒏
21) (53)6 = 518 22) (𝑛3)5 = 𝑛15
23) (22)6 = 212 24) [(−5)3]2 = (−5)6
Mixed Practice/Review:
25) 32 ∙ 3 ∙ 35 = 38 26) 610
65= 65
27) 𝑛9 ÷ 𝑛2 = 𝑛7 28) 58 × 52 = 510
29) (23)3 = 29 30) [(−4)2]9 = (−4)18
#31-36 Answers will vary
KEY
Name: _____________________________ Date:___________________
HW: Rules of Exponents
Fill in the blanks using expanded form to prove the answer is correct.
#1. 63 ∙ 62 = 65 #2. 76
74 = 72
(____ ∙ ____ ∙ ____) ∙ (____ ∙ ____) = 65
= 72
#𝟑. (32)4 = 38
32 ∙ ______ ∙ ______ ∙ ______ = _______
(____ ∙ ____) ∙ (____ ∙ ____) ∙ (____ ∙ ____) ∙ (____ ∙ ____) = 38
Fill in the blanks with the correct words. Then use the rule to simplify the example.
Rule #1 _______________________________________
When _______________exponents with the same base, 𝐄𝐱: 𝟗𝟕 ∙ 𝟗𝟐 = ______
____________ the exponents.
Rule #2 _______________________________________
When ________________ exponents with the same base, Ex: 𝟐𝟏𝟎
𝟐𝟒= _____
___________________ the exponents.
Rule #3 _______________________________________
When raising a ____________ to a ______________, Ex: [(−𝟓)𝟐]𝟏𝟑 ______
____________________ the exponents.
Practice
Simplify using exponential form for your answer.
1) 65 ∙ 65 =_______ 2) (−5)10
(−5)2= _________ 3)
83
81= _________
4) (73)4= _______ 5) (−4)4 ∙ (−4)3 =_______ 6) [(−2)4]4 = ________
Key- HW: Rules of Exponents
Fill in the blanks using expanded form to prove the answer is correct.
#1. 63 ∙ 62 = 65 #2. 76
74 = 72
(6 ∙ 6 ∙ 6) ∙ (6 ∙ 6) = 65 7∙7 ∙7 ∙7∙7∙7
∙7 ∙7 ∙7∙7 = 72
#𝟑. (32)4 = 38
32 ∙ 32 ∙ 32 ∙ 32 = 38
(3 ∙ 3) ∙ (3 ∙ 3) ∙ (3 ∙ 3) ∙ (3 ∙ 3) = 38
Fill in the blanks with the correct words. Then use the rule to simplify the example.
Rule #1 Product of Powers
When multiplying exponents with the same base, 𝐄𝐱: 𝟗𝟕 ∙ 𝟗𝟐 = 𝟗𝟗
add the exponents.
Rule #2 Quotient of Powers
When dividing exponents with the same base, Ex: 𝟐𝟏𝟎
𝟐𝟒 = 𝟐𝟔
subtract the exponents.
Rule #3 Power of a Power
When raising a power to a power, Ex: [(−𝟓)𝟐]𝟏𝟑 = (−𝟓)𝟐𝟔
multiply the exponents.
Practice
Simplify using exponential form for your answer.
1) 𝟔𝟓 ∙ 𝟔𝟓 = 𝟔𝟏𝟎 2) (−𝟓)𝟏𝟎
(−𝟓)𝟐= (−𝟓)𝟖 3)
𝟖𝟑
𝟖𝟏= 𝟖𝟐
4) (𝟕𝟑)𝟒= 𝟕𝟏𝟐 5) (−𝟒)𝟒 ∙ (−𝟒)𝟑 = (−𝟒)𝟕 6) [(−𝟐)𝟒]𝟒 = (−𝟐)𝟏𝟔
KEY
Simplify Exponential Expressions Partner Check
Directions: Cut the paper on the vertical line. Each partner gets one side. Complete your problem by simplifying the expression using exponents. Then stop, trade papers, and check your partner’s answer. Help each other correct any mistakes and then continue the process again until all problems are complete.
Name: ______________________________ Name: _______________________________
1. 34 ∙ 3 ∙ 33 2. (−5)6 ∙ (−5)4
STOP
3. [(−7)5]8 4. (26)4
STOP
5. 69
64 ∙ 63 6.
33 ∙ 34
3
STOP
7. (23)6 × (22)2
8. (83)4 × (82)5
STOP
9. 411
47× 410
10. 65
64× 62
STOP
11. (25)
4
(24)3 12. (74)
2
73
STOP
13. The Robinson toy company sold 108 boxes of Silly Bands. Each box of Silly Bands contains 104 Silly Bands. How many Silly Bands did they sell?
14. A warehouse contains 65 containers. Each container weighs 63 pounds. How much is the weight inside all of the containers?
STOP
15. A farm has 84 pigs. If all of the pigs weigh a total of 86 pounds, what is the average weight of each pig?
16. Jim wrote 107 words during a period of 103 days. On average, how many words did Jim write per day? STOP
KEY: Simplify Exponential Expressions Partner Check
Directions: Cut the paper on the vertical line. Each partner gets one side. Complete your problem by simplifying the expression using exponents. Then stop, trade papers, check your partner’s answer. Help each other correct any mistakes and then continue the process again until all problems are complete.
1. 34 ∙ 3 ∙ 33 𝟑𝟖
2. (−5)6 ∙ (−5)4 (−𝟓)𝟏𝟎
STOP
3. [(−7)5]8 (−𝟕)𝟒𝟎 4. (26)4 𝟐𝟐𝟒
STOP
5. 69
64 ∙ 63 𝟔𝟐 6.
33 ∙ 34
3 𝟑𝟔
STOP
7. (23)6 × (22)2 𝟐𝟐𝟐
8. (83)4 × (82)5 𝟖𝟐𝟐
STOP
9. 411
47× 410 𝟒𝟏𝟒
10. 65
64× 62 𝟔𝟑
STOP
11. (25)
4
(24)3 𝟐𝟖 12. (74)
2
73 𝟕𝟓
STOP
13. The Robinson toy company sold 108 boxes of Silly Bands. Each box of Silly Bands contains 104 Silly Bands. How many Silly Bands did they sell?
𝟏𝟎𝟏𝟐 Silly Bands
14. A warehouse contains 65 containers. Each container weighs 63 pounds. How much is the weight inside all of the containers?
𝟔𝟖 pounds STOP
15. A farm has 84 pigs. If all of the pigs weigh a total of 86 pounds, what is the average weight of each pig?
𝟖𝟐 pounds
16. Jim wrote 107 words during a period of 103 days. On average, how many words did Jim write per day?
𝟏𝟎𝟒 words per day STOP
KEY
I Have, Who Has?
Use this I have, Who Has game to review Exponents and Exponential Rules.
Product of Powers
Product of Quotients
Power of Power
Rules:
The game can be played with either whole class or with a small group.
First you must cut out the cards. Then pass out the cards to all members of the
group. It does not matter if they place them facing up or down. Start with any
card placed in the center. (If playing with the whole class, place the card on the
overhead projector.)
Start with any card. Read out the “Who Has____?” on the first card. Each player
looks to see if they have the corresponding answer. Whoever has it places it beside
the first card and reads out “I have____” and then “Who has____?” on that card.
Play continues until the last card is read. It will match the first card that was
played.
Game Variation-Pairs Dominos:
Each pair of students gets a set of cards. One player shuffles and deals out the
cards. The dealer picks one of her cards face up on the corner of her desk. Both
players then search through their hand to see if he/she has the card that is the
answer to the first card. Whomever has the next card, then places it either next
to or under the first card. Now both players look through their hand to search for
the answer to the problem on the 2nd card. This continues until all cards are played.
The winner could be whomever runs out of cards first (pure luck), or both players
win if they have a correct train of dominos on their desk.
Game Variation-Scavenger Hunt:
Place the cards around the room. Students can start anywhere and then find the
answer on the next card until they reach back to the beginning.
I have 𝟓𝟓
Who has? 𝟑𝟑
𝟑𝟏
I have −𝟐𝟕
Who has? – (𝟑)𝟒
I have 𝟑𝟐
Who has?
(𝟓𝟑)𝟑
I have -81
Who has?
6 squared
I have 𝟓𝟗
Who has? 𝟕𝟐 𝐱 𝟕
I have 𝟔𝟐
Who has? (−𝟑)𝟐 ∙ (−𝟑)𝟒 ∙ (−𝟑)
I have 𝟕𝟑
Who has? (−𝟖)𝟐 ∙ (−𝟖)𝟔
I have (−𝟑)𝟕
Who has? 𝟔𝟓 ÷ 𝟔𝟓
I have (−𝟖)𝟖
Who has? 𝟖𝟓 ÷ 𝟖𝟐
I have 𝟔𝟎
Who has? (𝟔𝟒)𝟒
I have 𝟖𝟑
Who has? 𝟐𝟖
𝟐𝟒
I have 𝟔𝟏𝟔
Who has? (−𝟗)𝟔
(−𝟗)𝟒
I have 𝟐𝟒
Who has? [(−𝟑)𝟓]𝟒
I have (−𝟗)𝟐
Who has? (−𝟑)𝟓 𝐱 (−𝟑)𝟒
I have (– 𝟑)𝟐𝟎
Who has? 𝟒𝟐 ∙ 𝟒𝟑
I have (−𝟑)𝟗
Who has? 𝟏𝟏𝟒
𝟏𝟕
I have 𝟒𝟓
Who has?
9 cubed
I have 𝟏𝟕
Who has? 𝟐𝟒 𝐱 𝟐𝟓
𝟐𝟑
I have 𝟗𝟑
Who has? 𝟑 ∙ 𝟑 ∙ 𝟑 ∙ 𝟑 ∙ 𝟑
I have 𝟐𝟔
Who has? 𝟐𝟏
I have 𝟑𝟓
Who has? 𝟓𝟕
𝟓
I have 𝟐
Who has? 𝟒𝟏𝟐
𝟒𝟒
I have 𝟓𝟔
Who has? 𝟒𝟐 ∙ 𝟒
I have 𝟒𝟖
Who has? (−𝟑)𝟒
I have 𝟒𝟑
Who has? 𝟏𝟗 𝐱 𝟏𝟑
I have 𝟖𝟏
Who has? 𝟕𝟔 𝐱 𝟕𝟔𝐱 𝟕𝟔
I have 𝟏𝟏𝟐
Who has? 𝟐𝟏𝟐
𝟐𝟓
I have 𝟕𝟏𝟖
Who has? 𝟓𝟑
I have 𝟐𝟕
Who has? (– 𝟑)𝟑
I have 𝟏𝟐𝟓
Who has? 𝟓𝟐 ∙ 𝟓𝟑
Connect 4: Exponential Rules Game Board
You and a partner will take turns choosing and giving an answer using exponential form. Ex. ( 𝑎𝑏) If
you are correct, you will place a chip on the block. If incorrect, your partner can steal the block if
he/she simplifies it correctly. If you disagree about the answer, ask your teacher. First player to get
four in a row, wins.
Materials needed: Game board, chips of 2 different colors.
95 ÷ 9
[(−9)2]²
(−6)9
(−6)3
710
• 76
76
• 76
45 • 47
210 x 25
28
÷ 26
(43)4•(42)2
𝑎8 • 𝑎2
(32)6
34
77
75
95 x 92
90
𝑥14
𝑥2
𝑥10 • 𝑥2
𝑥4 • 𝑥3• 𝑥6
23
• 27
24
• 25
(𝑎2)6
(𝑥2)3 • 𝑥6
126 • 123
𝑏5 ÷ 𝑏3
46
43
56x 57
54
75 • 79
16
• 17
13
69
60
(43)4• 47
75 ÷ 75
218 ÷ 25
24
• 26
[(−1)6]²
33
32
28• 22• 22
310
• 32
34
• 36
(−8)10
(−8)2
513• 52• 5
59 x 51
(52)4
82 • (82)4
610
68
x8y7 x6y7
Key-Connect 4: Exponential Rules
You and a partner will take turns choosing and giving an answer using exponential form. Ex. ( 𝑎𝑏) If you are
correct, you will place a chip on the block. If incorrect, your partner can steal the block if he/she simplifies it
correctly. If you disagree about the answer, ask your teacher. First player to get four in a row, wins.
Materials needed: Game board, chips of 2 different colors.
95 ÷ 9
94
[(−9)2]²
(−9)4
(−6)9
(−6)3
(−6)6
710
• 76
76
• 76
74
45 • 47
412
210 x 25
28
÷ 26
213
(43)4•(42)2
416
𝑎8 • 𝑎2
𝑎10
(32)6
34
38
77
75
72
95 x 92
90
97
𝑥14
𝑥2
𝑥12
𝑥10 • 𝑥2
𝑥4 • 𝑥3• 𝑥6
𝑥11
23
• 27
24
• 25
21
(𝑎2)6
𝑎12
(𝑥2)3 • 𝑥6
𝑥12
126 • 123
129
𝑏5 ÷ 𝑏3
𝑏2
46
43
43
56x 57
54
59
75 • 79
714
16
• 17
13
110
69
60
69
(43)4• 47
419
75 ÷ 75
70
218 ÷ 25
24
• 26
23
[(−1)6]²
(−1)12
33
32
31
28• 22• 22
212
310
• 32
34
• 36
32
(−8)10
(−8)2
(−8)8
513• 52• 5
516
59 x 51
(52)4
52
82 • (82)4
810
610
68
62
x8y7 x6y7
x14y14
KEY
Name:_________________________________________ Date:___________________________
Exponent Test
Fill in the blanks 1) When multiplying exponents with the same base, you ___________ the exponents.
2) When __________________ exponents with the same base, you subtract the exponents.
3) When finding the power of a power, you _____________________ the exponents.
Simplify #4-9 completely. 4) (−3)4 = ________ 5) −34 = ________ 6) (−20) 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 = ________
7) (−2)3 = ________ 8) (−5) 𝑐𝑢𝑏𝑒𝑑 = ________ 9) −23 = ___________
In #10-24, use your exponent rules to provide an answer in EXPONENT form.
22) Simplify: 24∙26
25 = ___________________
23) Simplify: 𝑥4∙𝑥6
𝑥3∙𝑥3 = _____________________
24) Simplify: 24 ∙ 26 ∙ (23)3 = ____________
10) 25 24 Answer: ___________
11) [(–3)5]3 Answer: ___________
12) (-12)4 (-12)6 Answer: ___________
13) 25017 250 Answer: ___________
14) 810 82 Answer: ___________
15) (z3)6 Answer: ___________
16) y20 y5 Answer: ___________
17) 137 13 133 Answer: ___________
18) (92)7
Answer: ___________
19) (–7)3 (–7)3 Answer: ___________
20) x15 x6 Answer: ___________
21) (656)6 Answer: ___________
Applications of Exponents Write the mathematical expression (Ex: 𝟐𝟑 ∙ 𝟐𝟓) and the answer with exponents and also in standard form including units. 25) A chicken plant receives 56 chickens daily. Each of the company trucks drop off 54
chickens once a day. How many trucks drop off chickens during the day? 26) Each large bag of potato chips has 122 chips in the bag. The grocery store
has 123 bags of potato chips in the store. How many potato chips are in the store all together?
27) The walk-in closets at a condo are cubes (meaning the length,
width, and height of the room are equivalent). The length of the closet is 8 feet. What is the volume of the closet? (Reminder: volume = length width height or s3)
28) A park has a fenced in area for people to let their dogs run
around without a leash. The enclosure is a square with side length 40ft. What is the area of the fenced in region?
*BONUS QUESTION*
Simplify this expression: Simplify: –4x5y4z 5xy7z4= ___________________
Key: Exponent Test Fill in the blanks 1) When multiplying exponents with the same base, you add the exponents.
2) When dividing exponents with the same base, you subtract the exponents.
3) When finding the power of a power, you multiply the exponents.
Simplify #4-9 completely. 4) (−3)4 = 81 5) −34 = −81 6) (−20) 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 = 400
7) (−2)3 = −8 8) (−5)𝑐𝑢𝑏𝑒𝑑 = −125 9) −23 = -8
In #10-24, use your exponent rules to provide an answer in EXPONENT form.
22) Simplify: 24∙26
25 = 25
23) Simplify: 𝑥4∙𝑥6
𝑥3∙𝑥3 = 𝑥4
24) Simplify: 24 ∙ 26 ∙ (23)3 = 219
10) 25 24 Answer: 29
11) [(–3)5]3 Answer: (-3)15
12) (-12)4 (-12)6 Answer: (-12)10
13) 25017 250 Answer: 25016
14) 810 82 Answer: 88
15) (z3)6 Answer: z18
16) y20 y5 Answer: y15
17) 137 13 133 Answer: 1311
18) (92)7
Answer: 914
19) (–7)3 (–7)3 Answer: (-7)0
20) x15 x6 Answer: x21
21) (656)6 Answer: 6536
KEY
Applications of Exponents Write the mathematical expression (Ex: 𝟐𝟑 ∙ 𝟐𝟓) and the answer with exponents and also in standard form including units. 25) A chicken plant receives 56 chickens daily. Each of the company trucks drop off 54 chickens once a day. How many trucks drop off chickens each day? 56
54 = 52 = 25 trucks
26) Each large bag of potato chips has 122 chips in the bag. The grocery store has 123 bags of potato chips in the store. How many potato chips are in the store all together? 122 123 = 125 = 248,832 potato chips 27) The walk-in closets at a condo are cubes (meaning the length, width, and height of the room are equivalent). The length of the closet is 8 feet. What is the volume of the closet? (Reminder: volume = length width height or s3) 8 x 8 x 8 = 83 = 512 ft3 28) A park has a fenced in area for people to let their dogs run around without a leash. The enclosure is a square with side length 40ft. What is the area of the fenced in region? 40 x 40 = 402 = 1,600 ft2
*BONUS QUESTION*
Simplify this expression: Simplify: –4x5y4z 5xy7z4= –20x6y11z5
KEY
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