Exponent Calculator
Calculate any base raised to any exponent including negative and fractional powers.
Includes a table of common squares, cubes, and powers of 2 and 10.
An exponent (or power) tells you how many times to multiply a base number by itself. The expression 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The base is 2, the exponent is 5.
Key exponent rules:
| Rule | Formula | Example |
|---|---|---|
| Product rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| Power rule | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional exponent | a^(1/n) = nth root of a | 8^(1/3) = 2 |
Worked examples:
Compound interest — money doubling: £1,000 at 7% annual interest for 10 years: £1,000 × 1.07¹⁰ = £1,000 × 1.9672 = £1,967.15
Population growth: A city of 500,000 grows at 2% per year for 20 years: 500,000 × 1.02²⁰ = 500,000 × 1.4859 = 742,974
Bacteria doubling every hour: Starting with 100 bacteria after 8 hours: 100 × 2⁸ = 100 × 256 = 25,600 bacteria
Scientific notation: Large numbers use exponents of 10. The distance from Earth to the Sun is approximately 1.496 × 10¹¹ metres. An electron’s mass is 9.109 × 10⁻³¹ kilograms.
Square and cube roots as fractional exponents: √16 = 16^(1/2) = 4 ∛27 = 27^(1/3) = 3
Exponents are fundamental to algebra, finance (compound growth), physics (exponential decay), and computer science (binary powers of 2).