An exponent tells you how many times to multiply a base by itself: 2⁴ = 2×2×2×2 = 16. A handful of rules cover almost all exponent work.
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | x² × x³ = x⁵ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁵ ÷ x² = x³ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
The pattern to remember: when you multiply same-base powers you add exponents; when you divide you subtract; a power of a power multiplies them. A zero exponent is always 1, and a negative exponent means a reciprocal.
Frequently asked questions
What is anything to the power of 0? 1.
What does a negative exponent mean? A reciprocal — a⁻ⁿ = 1/aⁿ.
How do I multiply powers? Same base: add the exponents.
Exponents are also how scientific notation and compound growth work, so the rules here show up far beyond algebra class — in everything from interest calculations to measuring the brightness of stars. Master the five rules and the rest builds on them.
A worked example ties the rules together: simplify (x³ × x⁴) ÷ x². Multiplying adds the exponents to give x⁷, then dividing subtracts to give x⁵. Negative and zero exponents follow naturally — x⁰ is 1, and x⁻² is 1/x² — so even a messy expression reduces in a few clean steps once you apply the rules in order.