Name: Date: Lesson 1.2 The Product and the Quotient of Powers
Use the product of powers property to simplify numerical expressions.
Example
a) 58  5
58  5 5
58 1 1
5
Use the product of powers property.
59
Simplify.
b) (22)3  (22)7
(22)3  (22)7 5 (22)
317
5
(22) 10
5
210
5
3
5
3
 2
 2
c)    
3
3
 2  2  3
 3 5
5
2 513
3
28
3
Use the product of powers property.
Simplify the exponent.
Simplify.
Use the product of powers property.
Simplify.
Complete.
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1. 134  139
134  139 5
Use the
Simplify.
5
of powers property.
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Name: Date: Complete.
2.(27)  (27)11
(27)  (27)11 5
Use the
5
Simplify the exponent.
5
Simplify.
of powers property.
3.0.92  0.94
0.92  0.94 5
Use the
Simplify.
5
of powers property.
4. 1810  186
5.(211)3  (211)4
6. 1.56  1.56
7.
6
 1  1
 5  5
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Simplify each expression. Write your answer in exponential notation.
10 Chapter 1 Lesson 1.2
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Name: Date: Use the product of powers property to simplify algebraic expressions.
Example
a) a2 ? a3
a2 ? a3 5
b) a2 1 3 5
a5
Use the product of powers property.
Simplify.
(2f )4 ? (2f )2
(2f )4 ? (2f ) 5 (2f )
412
5
5
c) Use the product of powers property.
(2f )6 f 6
Simplify the exponent.
Simplify.
(9p)10 ? (9p)10
(9p)10 ? (9p)10 5 (9p)
10 1 10
5
(9p)20 Use the product of powers property.
Simplify.
Complete.
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8.b  b6
b  b6 5
Use the
5
Simplify.
of powers property.
9.(2q)3  (2q)3
(2q)3  (2q)3 5
Use the
5
Simplify the exponent.
5
Simplify.
of powers property.
10.
(4s)8  (4s)2
(4s)8  (4s)2 5
Use the
Simplify.
5
of powers property.
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Name: Date: Simplify each expression. Write your answer in exponential notation.
11. w12 ? w15
12.(2k)41 ? (2k)
13.(10t)5 ? (10t)4
Use the product of powers property to simplify algebraic expressions.
Example
a) a2 b3 ? a3 b2
a2 b3 ? a3 b2 5 a2 ?
5 a2 ? a3 ?b 3 ? b 2
Regroup factors with the same base.
5 a2 1 3 ? b 3 1 2
Add the exponents of the factors with the same base.
5 a5b 5
Simplify.
b3 ?a3 ? b 2
Rewrite the product.
b) 4p2 q4 ? 2p4 q6
4p2 q4 ? 2p4 q6 5 4
? p 2 ? q4 ? 2 ? p 4 ? q6
5 4 ? 2 ? p 2 ? p 4 ? q4 ? q6
5 8 ? p 2 1 4 ? q4 1 6
5 8p 6q10
Rewrite the product.
Regroup numbers, and regroup factors with
the same bases.
Add the exponents of the factors with the same base.
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Simplify.
12 Chapter 1 Lesson 1.2
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Name: Date: Complete.
1
4.x2y ? x4y3
x2y ? x4y3 5
?
?
?
the product.
5
?
?
?
5
?
the exponents of the factors
5
egroup the factors with the
R
base.
with the same base.
Simplify.
Simplify each expression. Write your answer in exponential notation.
15. c4 d2  c5 d6 16. 2j3k5  3j2k7
Use the quotient of powers property to simplify numerical expressions.
Example
a) 35 4 32
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35 4 32 5
b) 35 2 2 5
33
Use the quotient of powers property.
Simplify.
(22)8 4 (22)4
(22)8 4 (22)4 5 (22)
824
5
5
c) (22)4 24
Use the quotient of powers property.
Simplify the exponent.
Simplify.
2.39 4 2.32
2.39 4 2.32 5
5
2.39 2 2 2.37 Use the quotient of powers property.
Simplify.
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Name: Date: Complete.
17. 619 4 614
619 4 614 5
Use the
Simplify.
5
of powers property.
18. (24)7 4 (24)3
(24)7 4 (24)3 5
Use the
5
Simplify the exponent.
5
Simplify.
1
19.  
 3
17
1
 
 3
11
 1
 3
17
1
 
 3
11
5
Use the
5
Simplify.
of powers property.
of powers property.
20. 715 4 77
21.(26)4 4 (26)3
22. 0.256 4 0.254
 4
 4
23.    
5
5
6
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Simplify each expression. Write your answer in exponential notation.
14 Chapter 1 Lesson 1.2
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Name: Date: Use the quotient of powers property to divide algebraic expressions.
Example
a) q4 4 q3
q4 4 q3 5
q 4 2 3
Use the quotient of powers property.
q
Simplify.
5
b)(2p)3 4 (2p)
(2p)3 4 (2p) 5 (2p)
321
Use the quotient of powers property.
5
(2p)2
Simplify the exponent.
5
p2
Simplify.
Complete.
2
4. x9 4 x7
x9 4 x7 5
Use the
Simplify.
5
of powers property.
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2
5. (2r)12 4 (2r)2
(2r)12 4 (2r)2 5
Use the
Simplify.
5
of powers property.
Simplify each expression. Write your answer in exponential notation.
2
6. y14 4 y11
27.(2a)10 4 (2a)4
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Name: Date: Use the quotient of powers property to simplify algebraic expressions.
Example
a) u5s6 4 u3s4
u5s6 4 u3s4 5
u 5s 6
u 3s 4
Write the quotient as a fraction.
5
u 5 s 6
? u 3 s 4
Rewrite the fraction as a product of two fractions.
5 u 523 ? s6 2 4
Use the quotient of powers property.
5 u 2s 2
Simplify.
b)16a6b2 4 8a3b
16a6b2 4 8a3b 5
16a6b2
8a3b
Write the quotient as a fraction.
16 a6 b 2
? ? 8 a3 b
Rewrite the fraction as a product of three fractions.
5
5 2 ? a6 2 3 ? b 2 2 1
Use the quotient of powers property.
5 2a3b
Simplify.
Complete.
2
8. e17f 8 4 e13f 4
5
?
ewrite the fraction as a
R
fractions.
5
?
Use the
5
Write the quotient as a
Simplify.
.
of two
of powers property.
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e17f 8 4 e13f 4 5
16 Chapter 1 Lesson 1.2
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Name: Date: Complete.
2
9. 81x4y17 4 27x3y12
81x4y17 4 27x3y12 5
5
?
Write the quotient as a
?
Rewrite the fraction as a
.
of three
fractions.
5
5
?
?
Use the
of powers property.
Simplify.
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Simplify each expression. Write your answer in exponential notation.
3
0. w 9v 6z 4 4 w 2v 3z 3
31.169d 10g 8 4 13d 7g 2
3
2. c 4m9p12 4 c 2m 5p 9
33.350k12m3 4 70k6m
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Name: Date: Use exponent properties to simplify numerical and algebraic expressions.
Example
35 38 33
32 35 3
a)
5
35 1 8 1 3
32 1 5 1 1
Use the product of powers property.
5
316
38
Simplify.
5 316 2 8
Use the quotient of powers property.
5 38
Simplify.
b)
35 38 33
32 35 3
3
4
5
3
4
5
 1  1  1
2  2  2 
4
3
2
 1  1  1
2  2  2 
3 4 5
 1  1  1
2  2  2 
4
3
2
 1  1  1
2  2  2 
1
5 2
1
2
1 12
5 2 9
1
2
4 3 2
Use the product of powers property.
Simplify.
12 9
5
1
2
5
13
2
Use the quotient of powers property.
Simplify.
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18 Chapter 1 Lesson 1.2
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Name: c)
2a 6 3a 3 4a 2
a 3 3a 4 5a 2
2a 6 3a 3 4a 2
3
a 3a 4 5a 2
2 · 3 · 4 · a6 · a3 · a2
5
1 · 3 · 5 · a3 · a4 · a2
5
24 a 6 3 2
15a 3 4 2
5
8a 11
Simplify.
5a 9
5
8 11 9
(a
) 5
Date: 5 8 a 2 5
egroup the numbers, and regroup the
R
factors with the same bases.
Use the product of powers property.
Use the quotient of powers property.
Simplify.
Complete.
88 82 8
34. 84 82 83
Use the
5
Simplify.
5
Use the
5
Simplify.
of powers property.
of powers property.
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88 82 8
5
8 4 82 83
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Name: Date: Complete.
35.
(0.35)8 (0.35)7 (0.35)3
(0.35)5 (0.35)2 (0.35)3
(0.35)8 (0.35)7 (0.35)3
5
(0.35)5 (0.35)2 (0.35)3
Use the
5
Simplify.
5
Use the
5
Simplify.
of powers property.
of powers property.
x 3 8x 2 18x 9
36. 2x 6 x 7 3x 5
x 3 8x 2 18x 9
5
2x 6 x 7 3x 5
egroup the numbers, and regroup the
R
factors with the same bases.
5
Use the
5
Simplify.
5
Use the
5
Simplify.
of powers property.
of powers property.
Simplify each expression. Write your answer in exponential notation.
37.
6
3
2
 
5 
3
2
 
5 
38.
2p 4 15p12 10p10
2p 3 3p 6 5p 4
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7
2  2 
5  5 
5
4
2  2 
5  5 
20 Chapter 1 Lesson 1.2
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Name: Date: Solve a real-world problem in exponential notation.
Example
Heather extracted 1 microliter of blood sample from a vile containing 100 microliters
of blood. The sample she extracted contained approximately 1026 red blood cells.
Predict how many red blood cells were initially in the vile of blood? Write your
answer in exponential notation.
Number of red blood cells in 1 vile
5 Number of red blood cells in 1 microliter of blood sample ? Number of liters of blood sample
5
1026
5
10
5
5
100
?
10
26
?
2
Substitute values.
Rewrite 100 as 102.
1026 1 2 1028
Use the product of powers property.
There were approximately
Simplify.
10
28
red blood cells in the vile of blood.
Solve. Show your work.
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3
9. A grain of sand has a mass of approximately 102 milligrams. A rock has a mass
of approximately 1010 milligrams. How many times more is the mass of the rock
than the mass of the grain of sand?
0. There are approximately 104 blueberries in one crate. In a particular week, a
4
blueberry farm sold 1,000 crates of blueberries. How many blueberries were
sold in that week? Write your answer in exponential notation.
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