Quotient Rule for Exponents
When dividing powers with the same base, subtract the exponents:
[
\frac{a^m}{a^n}=a^{m-n} \quad (a\neq 0)
]
This rule works because repeated factors cancel. The key is to keep the base the same and only combine the exponents.
Method
- Check the base. If the numerator and denominator have the same base, you can use the quotient rule.
- Subtract the exponents. Write the base once and compute top exponent minus bottom exponent.
- Simplify the result. If the exponent becomes 0, the answer is 1. If it is negative, rewrite if needed using a fraction.
Examples of what to do
- (x^7/x^3 = x^{7-3} = x^4)
- (a^5/a^8 = a^{5-8} = a^{-3})
If you see coefficients or different bases, simplify those parts separately before using the rule.
Check your work
Make sure:
- the bases are the same,
- you subtracted in the correct order,
- and the final expression is written in simplest form.
A good final check is to compare the sizes of the exponents: dividing by a larger power often gives a negative exponent or a fraction.