The laws of exponents
The rules hold for any nonzero base a, b and any exponents m and n (real a, b > 0 when the exponents are fractions or irrational).
| Rule | Formula | Example |
|---|---|---|
| Multiplying (same base) | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Dividing (same base) | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 8 × 125 = 1000 |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)² = 4/9 |
| Same exponent, multiply bases | aⁿ × bⁿ = (ab)ⁿ | 4² × 5² = 20² = 400 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 = 0.125 |
| Negative exponent of a fraction | (a/b)⁻ⁿ = (b/a)ⁿ | (2/5)⁻² = 25/4 |
| Fractional exponent | a^(1/n) = ⁿ√a | 8^(1/3) = 2 |
| Fractional exponent m/n | a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ | 27^(2/3) = 3² = 9 |
| One as exponent | a¹ = a | 9¹ = 9 |
Sources: OpenStax: College Algebra 2e §1.2 Exponents and Scientific Notation
Adding and subtracting exponents
There is no rule that combines aᵐ + aⁿ into a single power. Powers can be added only when they are like terms, the same base with the same exponent: 3 × 2⁴ + 5 × 2⁴ = 8 × 2⁴ = 2⁷.
With different exponents, factor out the smaller power: 2⁵ + 2³ = 2³(2² + 1) = 8 × 5 = 40. “Adding exponents” as a rule refers to multiplying powers of the same base, where the exponents add.
Simplifying expressions with exponents
Apply the rules in this order: expand powers of products and quotients, combine powers of the same base, then rewrite negative exponents as fractions.
- (x³y²)² ÷ (xy)³ = x⁶y⁴ ÷ x³y³ = x³y
- (2a⁻²b)³ = 8a⁻⁶b³ = 8b³ ÷ a⁶
- √(x⁵) × x^(1/2) = x^(5/2) × x^(1/2) = x³
- 0⁰ is left undefined in many algebra courses; in combinatorics, series and most software it is taken to be 1