NAME _____________________________________________ DATE ____________________________ PERIOD _____________
7-2 Study Guide and Intervention
Division Properties of Exponents
Divide Monomials To divide two powers with the same base, subtract the exponents.
𝑎𝑚
= 𝑎𝑚 − 𝑛 .
Quotient of Powers
For all integers m and n and any nonzero number a,
Power of a Quotient
For any integer m and any real numbers a and b, b ≠ 0, ( ) =
Example1: Simplify
𝒂𝟒 𝒃𝟕
.
𝒂𝒃𝟐
𝑎𝑛
Example 2: Simplify (
denominator equals zero.
𝑎4 𝑏7
𝑎𝑏2
(
=
= (𝑎
4−1
)( 𝑏
3
2𝑎 3 𝑏5
)
3𝑏2
Group powers with the same base.
7−2
3 5
=ab
𝑎𝑚
𝑏
𝑎𝑚
𝟑
𝟐𝒂𝟑 𝒃𝟓
)
.
𝟐
𝟑𝒃
Assume that no
denominator equals zero.
𝑎4
𝑏7
( 𝑎 ) (𝑏2 )
𝑎 𝑚
Assume that no
3
=
) Quotient of Powers
(2𝑎 3 𝑏5 )
(3𝑏2 )3
3
=
Simplify.
.
The quotient is 𝑎3 𝑏 5
=
=
23 (𝑎 3) (𝑏5)
(3)3 (𝑏
Power of a Quotient
3
2) 3
8𝑎 9 𝑏15
27𝑏6
8𝑎9 𝑏9
27
The quotient is
Power of a Product
Power of a Power
Quotient of Powers
8𝑎 9 𝑏9
.
27
Exercises
Simplify each expression. Assume that no denominator equals zero.
55
𝑚6
1. 52
2. 𝑚4
3.
4.
𝑎2
𝑎
5. 𝑥 5 𝑦2
7.
𝑥𝑦 6
𝑦4𝑥
8. (
2𝑟 5 𝑤 3
𝑥 5𝑦3
Chapter 7
−2y7
6. 14y5
3
2𝑎2 𝑏
)
𝑎
4
10. ( 𝑟4 𝑤 3 )
3𝑟 6 𝑛3
3
4𝑝4 𝑟 4
)
2
2
3𝑝 𝑟
9. (
4
11. ( 2𝑟5 𝑛 )
11
𝑝5 𝑛 4
𝑝2 𝑛
𝑟 7 𝑛7 𝑡 2
12. 𝑛3 𝑟3 𝑡 2
Glencoe Algebra 1
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
7-2 Study Guide and Intervention (continued)
Division Properties of Exponents
Negative Exponents Any nonzero number raised to the zero power is 1; for example, (−0.5)0 = 1. Any nonzero
number raised to a negative power is equal to the reciprocal of the number raised to the opposite power; for example,
1
6−3 = 3 . These definitions can be used to simplify expressions that have negative exponents..
6
Zero Exponent
For any nonzero number a, 𝑎0 = 1.
Negative Exponent Property
For any nonzero number a and any integer n, 𝑎−𝑛 =
1
𝑎𝑛
n and
1
𝑎−𝑛
= 𝑎𝑛 .
The simplified form of an expression containing negative exponents must contain only positive exponents.
𝟒𝒂−𝟑 𝒃𝟔
Example: Simplify 𝟏𝟔𝒂𝟐𝒃𝟔𝒄−𝟓. Assume that no denominator equals zero.
4𝑎 −3 𝑏 6
16𝑎2 𝑏6 𝑐 −5
𝑎 −3
4
𝑏6
1
= (16) ( 𝑎2 ) (𝑏6 ) (𝑐 −5 )
1
= 4 (𝑎−3 − 2 )( 𝑏 6 − 6)(𝑐 5 )
1
= 4 𝑎−5 𝑏 0 𝑐 5
1
1
= 4 (𝑎5 )(1)𝑐 5
=
𝑐5
4𝑎5
Group powers with the same base.
Quotient of Powers and Negative Exponent Properties
Simplify.
Negative Exponent and Zero Exponent Properties
Simplify.
𝑐5
The solution is 4𝑎5.
Exercises
Simplify each expression. Assume that no denominator equals zero.
22
1. 2−3
𝑏−4
4. 𝑏−5
7.
𝑥4𝑦0
𝑥 −2
𝑚−3 𝑡 −3
𝑡 3 )−1
10. (𝑚2
Chapter 7
𝑚
2. 𝑚−4
3.
𝑝−8
𝑝3
6.
(𝑎 2 𝑏3 )
(𝑎𝑏)−2
0
5.
2
(−𝑥 −1 )
4𝑤 −1 𝑦 2
2
8.
(6𝑎−1 𝑏)
(𝑏2 )4
9.
0
4𝑚2 𝑛2
)
−1
8𝑚 ℓ
11. (
12
(3𝑟𝑡)2 𝑢−4
𝑟 −1 𝑡 2 𝑢7
−3
12.
(−2𝑚𝑛2 )
4𝑚−6 𝑛4
Glencoe Algebra 1