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Integer Exponents and Their Laws

Algebra Foundation

Algebra 1

Study expressions, equations, inequalities, functions, linear models, systems, polynomials, quadratics, and exponential growth with original explanations and worked examples.

Learn with original explanations and worked examples. OpenStax Elementary Algebra 2e was consulted for topic coverage.

Study every current unit through original concept lessons, formula derivations, experiments, quizzes, and practice sets.

Algebra 1 · Unit 7 · Topic 7.1

Integer Exponents and Their Laws

For a nonzero base a and integer exponents, aᵐaⁿ = aᵐ⁺ⁿ and aᵐ/aⁿ = aᵐ⁻ⁿ.

Definition and explanation

For a nonzero base a and integer exponents, aᵐaⁿ = aᵐ⁺ⁿ and aᵐ/aⁿ = aᵐ⁻ⁿ. A power raised to an integer power satisfies (aᵐ)ⁿ = aᵐⁿ. Nonzero restrictions matter when division or nonpositive powers occur.

Worked example

For x ≠ 0, simplify (x³x⁵)/x². Add numerator exponents and subtract the denominator exponent: x⁸/x² = x⁶. For x = 2, both the original and simplified expressions equal 64.

Common mistake

The exponent rule for multiplying powers does not apply to adding powers.

Key takeaway

For a nonzero base a and integer exponents, aᵐaⁿ = aᵐ⁺ⁿ and aᵐ/aⁿ = aᵐ⁻ⁿ.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Elementary Algebra 2e and Algebra and Trigonometry 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.