Quotient of Powers
Divide powers with the same base by subtracting exponents.
Formula
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Example
x^5/x^2 = x^3
How it works
When two powers have the same base, dividing them means canceling matching factors. For example, a^m contains m copies of a multiplied together and a^n contains n copies of a; if you divide, the copies that appear in both numerator and denominator cancel, leaving m - n copies of a. This is why the exponents subtract, as long as the base is the same and the base is not zero.
When to use it
Use this rule whenever you simplify algebraic fractions with the same base, such as in exponent problems, scientific notation, and expressions that come from factoring. It is especially helpful for reducing expressions before solving equations.
Worked example, step by step
Simplify 7^6 / 7^2 7^6 / 7^2 = 7^(6-2) 7^(6-2) = 7^4 7^4 = 2401 Final answer: 2401
Common mistakes to avoid
- Adding the exponents instead of subtracting them
- Using the rule when the bases are different
- Forgetting that the formula only works for division, not multiplication
Did You Know?
Dividing powers means subtracting exponents, the mirror image of the rule for multiplying them.
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