Quotient of Powers
Dividing powers with the same base subtracts exponents.
Formula
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Example
2⁷ / 2³ = 2⁴ = 16.
How it works
When you divide two powers that have the same base, you can cancel matching factors. For example, a^m means a multiplied by itself m times, and a^n means a multiplied by itself n times; dividing them removes n copies of a from the top, leaving a^(m-n). This rule works for any nonzero base a, and it is one of the main exponent laws used to simplify algebraic expressions.
When to use it
Use this rule when simplifying fractions with repeated factors in the numerator and denominator, especially in algebra problems with variables or scientific notation. It is also useful when comparing or simplifying expressions before solving equations.
Worked example, step by step
Simplify 5^6 / 5^2 Use the quotient of powers rule: 5^(6-2) Subtract the exponents: 5^4 Compute the value: 5^4 = 625 Final answer: 625
Common mistakes to avoid
- Subtracting the bases instead of the exponents
- Using this rule when the bases are different
- Forgetting that the base must be nonzero
Did You Know?
Subtracting exponents when dividing powers naturally leads to the definition of negative and zero exponents.
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