\frac{a^m}{a^n}=a^{m-n}Variables
- a: nonzero base
- m,n: exponents
How to use this formula
Divides powers with the same nonzero base by subtracting exponents.
Important notes
- a cannot be zero when negative exponents may result.
Quick example
x⁷/x²=x⁵.
Applicability, worked calculation, and verification
Domain and applicability
Apply Exponent Quotient Rule only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- a cannot be zero when negative exponents may result.
- For the Exponent Quotient Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Exponent Quotient Rule, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Exponent Quotient Rule when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Exponent Quotient Rule, check zero, negative, and extreme input values before relying on the result.
- When using Exponent Quotient Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \frac{a^m}{a^n}=a^{m-n} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Divide 3⁷ by 3². Subtract the exponents to get 3⁵=243.
- Confirm the common nonzero base 3.
- Subtract exponents: 7-2=5.
- Evaluate 3⁵=243.
Independent verification
Direct cancellation of two factors of 3 leaves five factors, confirming 243.
Common mistakes
- When copying Exponent Quotient Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Exponent Quotient Rule, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Exponent Quotient Rule in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\frac{a^m}{a^n}=a^{m-n}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Exponent Quotient Rule.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Exponent Quotient Rule used for?
Divides powers with the same nonzero base by subtracting exponents.
Can I copy this formula as LaTeX?
Yes. Copy \frac{a^m}{a^n}=a^{m-n} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.