a^m a^n=a^{m+n}Variables
- a: common base
- m,n: exponents
How to use this formula
Combines powers with the same base by adding exponents.
Important notes
- The base must be the same.
Quick example
x³x⁵=x⁸.
Applicability, worked calculation, and verification
Domain and applicability
Apply Exponent Product 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
- The base must be the same.
- For the Exponent Product 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 Product Rule when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Exponent Product Rule, check zero, negative, and extreme input values before relying on the result.
- When using Exponent Product Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form a^m a^n=a^{m+n} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Multiply 2³·2⁵. Because the base is the same, add the exponents: 2⁸=256.
- Confirm that both powers have base 2.
- Add the exponents 3+5=8.
- Evaluate 2⁸=256.
Independent verification
Direct multiplication 8×32=256 confirms the result.
Common mistakes
- Before substituting values into Exponent Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Exponent Product 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 Product 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
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 Product 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 Product Rule used for?
Combines powers with the same base by adding exponents.
Can I copy this formula as LaTeX?
Yes. Copy 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.