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, near enough to doubling: $1,000 at 7% annual interest for 10 years: $1,000 × 1.07¹⁰ = $1,000 × 1.9672 = $1,967.15 Not quite double, which is the rule of 72 telling the truth: 72 ÷ 7 is 10.3 years, so you are three months short.
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
Two cases that trip people up. A negative base with a whole-number exponent is fine: (−2)³ = −8, and the sign of the answer just follows whether the exponent is odd or even. A negative base with a fractional exponent is not fine. (−8)^(1/2) asks for the square root of a negative number, which no real number provides, so this calculator reports that there is no real answer rather than inventing one. The cube root of −8 is genuinely −2 by hand, but computers refuse it too, because they cannot tell the fraction 1/3 from the decimal 0.3333333333 that you actually typed.
The other one is 0⁰. Most software, this page included, returns 1. That is a convention rather than a discovery, chosen because it makes the binomial theorem and power series come out right. Some texts leave it undefined and both camps have a point.
Exponents are fundamental to algebra, finance (compound growth), physics (exponential decay), and computer science (binary powers of 2).
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.