Expressions & Equations

Properties of Exponents

Applying the properties of integer exponents to generate equivalent expressions.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Rules for Combining Powers

Before reading on, predict: what should x³ × x² simplify to? Try writing both out as repeated multiplication first — x³ means x times itself 3 times, and x² means x times itself 2 times — and count how many total x's end up multiplied together.

Writing it out: x³ × x² = (x · x · x) × (x · x) = x · x · x · x · x, which is x⁵. Notice the exponents, 3 and 2, added together to give 5 — and that's not a coincidence specific to this example. Multiplying two powers of the same base always combines their total count of factors, which is exactly why exponents add when powers with the same base are multiplied.

Definition — Properties of Exponents

For a nonzero base a and integer exponents m and n: the product rule states aᵐ · aⁿ = aᵐ⁺ⁿ; the power rule states (aᵐ)ⁿ = aᵐⁿ; the quotient rule states aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Any nonzero number raised to the power 0 equals 1, and a⁻ⁿ equals 1/aⁿ.

The zero and negative exponent rules follow the same logic as extending a pattern, the way negative-number multiplication did in Grade 7. Following the quotient rule's pattern downward: x³÷x³ = x⁰, but also x³÷x³ obviously equals 1 by direct division — so x⁰ must equal 1. Continuing the pattern to x⁻¹, x⁻² keeps dividing by another factor of x each time, landing on fractions: x⁻¹ = 1/x.

Worked Example — Applying the Product and Power Rules

Simplify (2x³)² × x⁴. First apply the power rule to (2x³)²: 2² × x⁶ = 4x⁶. Then apply the product rule: 4x⁶ × x⁴ = 4x¹⁰.

Worked Example — Applying the Quotient Rule and Negative Exponents

Simplify x³ ÷ x⁵. By the quotient rule: x³⁻⁵ = x⁻². Rewriting with a positive exponent: x⁻² = 1/x².

a^{m} \cdot a^{n} = a^{m+n} \qquad (a^{m})^{n} = a^{mn} \qquad a^{-n} = \dfrac{1}{a^{n}}

Tip

The product and quotient rules only combine exponents directly when the bases already match — x³ · y² has no shortcut simplification, since x and y are different bases.

Common Mistakes

Multiplying the exponents instead of adding them when multiplying two powers of the same base, such as simplifying x³ · x² as x⁶.
Exponents add when multiplying same-base powers (the product rule) and multiply only when raising a power to another power (the power rule) — these are two different rules for two different situations.
Leaving a final answer with a negative exponent when a positive-exponent form was expected, or mishandling the reciprocal when rewriting it.
A negative exponent means 'reciprocal of the positive-exponent version' — a⁻ⁿ becomes 1/aⁿ, moving the base (and only the base) to the denominator.

Key Takeaways

Exponents add when multiplying same-base powers, and multiply when raising a power to another power.
Any nonzero number to the power 0 equals 1, following naturally from extending the quotient rule's pattern.
A negative exponent represents a reciprocal: a⁻ⁿ = 1/aⁿ.

Summary

These exponent rules, grounded in counting repeated factors, will be used constantly through Algebra I. The next lesson uses exponents in reverse, asking what number produces a given power in the first place.

Track your progress

Back to Expressions & Equations Square & Cube Roots→