7-2 Division Properties of Exponents
Students will be able to: use the structure of an expression to identify ways to rewrite it and to use the properties of
exponents to interpret expressions for exponential functions.
Key Words
Any nonzero number raised to the zero power
Any nonzero real number raised to a negative
power
The number rounded to the nearest power of 10
Zero Exponent
Negative Exponent
Order of Magnitude
Quotient of Powers
Our goal is: to divide two powers with the same base by subtracting the exponents
Symbolically:
𝑎𝑚
𝑎𝑛
28
Examples: 3 =
2
= am-n
25
Example 1
Simplify each expression. Assume that no denominator equals 0.
𝑔 3 ℎ5
𝑔ℎ2
Step 1: Subtract your exponents by grouping powers with the same base
g3-1
h5-2
Step 2: Simplify
g2h3
𝑥3𝑦4
𝑥2𝑦
X3-2 y4-1
xy3
Power of a Quotient
Our goal is: to find the power of a quotient by find the power of the numerator and the power of the denominator
Symbolically:
𝒂
𝒂𝒎
𝒃
𝒃𝒎
( )𝒎 =
𝟑 𝟓
Examples: ( )
𝟒
=
𝟑𝟓
𝟒𝟓
Example 2
Simplify each expression.
3𝑥 4 3
)
4
3𝑝3 2
)
7
(
(
Step 1: The power goes to the coefficient and the variable
32 (𝑝3 )2
72
Step 2: Simplify using power of a product from
yesterday’s lesson
9𝑝6
49
33 (𝑥 4 )3
43
27𝑥 12
64
Zero Exponent Property
Our goal is: any nonzero number raised to the zero power is equal to 1
Symbols: (b)0 = 1
Example: (15)0 = 1
Negative Exponents
Our goal is: for any nonzero number a and any integer n, 𝑎−𝑛 is the reciprocal of an, likewise the reciprocal of 𝑎−𝑛 is an
Symbols: 𝑎−𝑛 =
1
𝑎𝑛
Example: 2−4 =
1
24
1
= 16
Example 3
Simplify each expression. Assume that no denominator equals zero.
𝒏−𝟓 𝒑𝟒
𝒓−𝟐
Step 1: Rewrite the expression with all
positive exponents.
𝟏
𝒓𝟐
𝟒
∗
𝒑
∗
𝟏
𝒏𝟓
Step 2: Simplify each term
𝟏
𝒓𝟐
𝟒
∗
𝒑
∗
𝟏
𝒏𝟓
Step 3: Combine terms using multiplication
𝒑𝟒 𝒓𝟐
𝒏𝟓
𝟓𝒓−𝟑 𝒕𝟒
−𝟐𝟎𝒓𝟐 𝒕𝟕 𝒖−𝟓
𝟓
𝟏
𝒕𝟒 𝒖𝟓
∗ 𝟐
∗
∗
−𝟐𝟎 𝒓 ∗ 𝒓𝟑 𝒕𝟕 𝟏
𝟐𝒂𝟐 𝒃𝟑 𝒄−𝟓
𝟏𝟎𝒂−𝟑 𝒃−𝟏 𝒄−𝟒
𝟐 𝒂𝟐 ∗ 𝒂𝟑 𝒃𝟑 ∗ 𝒃𝟏 𝒄𝟒
∗
∗
∗ 𝟓
𝟏𝟎
𝟏
𝟏
𝒄
𝟏
𝟏 𝟏 𝒖𝟓
∗ 𝟓∗ 𝟑∗
−𝟒 𝒓 𝒕
𝟏
𝟏 𝒂𝟓 𝒃𝟒 𝟏
∗
∗
∗
𝟓 𝟏 𝟏 𝒄
𝒖𝟓
−𝟒𝒓𝟓 𝒕𝟑
𝒂𝟓 𝒃𝟒
𝟓𝒄