1. Quotient Rules
Integrated Math II 10.3

2. Negative Exponents 𝑋 −𝑛 = 1 𝑋 𝑛 Example: Simplify 𝑥 −3 = 1 𝑥 3
Negative exponents mean division.
𝑋 −𝑛 = 1 𝑋 𝑛
Example: Simplify 𝑥 −3 = 1 𝑥 3
Note: Simplify a single term with exponents means to make it so that each variable only appears once and that all exponents are positive. Remember “terms” do not have addition or subtraction

3. The Quotient Rule 𝑋 𝑚 𝑋 𝑛 = 𝑋 𝑚−𝑛
To divide two numbers with the same base, subtract the exponents.
𝑋 𝑚 𝑋 𝑛 = 𝑋 𝑚−𝑛
Example: Simplify 𝒙 𝟕 𝒙 𝟒 = 𝒙 𝟑
Note: For exponents, division is subtraction.

4. The Exponent “0” Anything raised to the power “0” is 1 𝑋 0 =1
Example: Simplify 5,287,613 𝑥 50 𝑦 99 𝑎 13 𝑏 −15 𝑧 = 1
Note: It is generally a good idea to look for “0” exponents so you can quickly simplify these terms and factors.

5. Exponents of Quotients
The power of a quotient, is the quotient of the powers
𝑋 𝑌 𝑛 = 𝑋 𝑛 𝑌 𝑛
Example: Simplify = =
Note: We know this fraction cannot be simplified further.
The numerator only has 2 for factors.
The denominator only has 5 for factors.

6. Bringing It Together
Break the problem down, then handle each set of numbers and each variable in turn.
Example: 𝟔 𝒂 −𝟑 𝒃 𝟕 𝒄 −𝟒 𝟑 𝒂 𝟔 𝒃 𝟐 𝒄 −𝟐 = 𝟔 𝟑 𝒂 −𝟑 𝒂 𝟔 𝒃 𝟕 𝒃 𝟐 𝒄 −𝟒 𝒄 −𝟐
=2 𝑎 −3−6 𝑏 7−2 𝑐 −4−(−2) =2 𝑎 −9 𝑏 5 𝑐 −2
= 𝟐 𝑏 5 𝑎 9 𝑐 2

7. Division and Scientific Notation
Group the numbers and the 10’s just like before
Put it back in Scientific Notation
Example: Simplify 2 × × 10 3
= ×
= 0.25 × 10 4
= 2.5 × 10 3

8. Summary Quotient Rule Negative exponents Exponents of quotients
The exponent “0”
Division in scientific notation
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