How this works
An exponent (also called a power) tells you how many times to multiply a number by itself. 2³ means 2 × 2 × 2 = 8. The base is the number being multiplied; the exponent is how many copies of it. This generalises far beyond positive whole-number exponents: negative exponents give reciprocals (2⁻³ = 1/8), zero gives 1 (any non-zero base⁰ = 1), and fractional exponents give roots (2^(1/2) = √2 ≈ 1.414, the square root; 8^(1/3) = ∛8 = 2, the cube root). Decimal exponents work too: 2^2.5 = 2² · 2^0.5 = 4 · √2 ≈ 5.657.
The rules of exponents make algebraic manipulation systematic. Same base, multiplied: add exponents (2³ · 2⁴ = 2⁷). Same base, divided: subtract exponents (2⁷ / 2³ = 2⁴). Power of a power: multiply exponents ((2³)² = 2⁶). Same exponent across different bases: factor or distribute ((ab)² = a² · b²; a²/b² = (a/b)²). These rules let you simplify expressions like 8^(2/3) without a calculator: 8^(2/3) = (8^(1/3))² = 2² = 4.
A few practical points. (1) For very large or very small results, this calculator switches to scientific notation (1.234 × 10²³ instead of a 24-digit number) since most contexts can't use raw numbers that big. (2) Negative bases with non-integer exponents are tricky — (-8)^(1/3) is technically -2 (real cube root) but the same input via complex-number rules gives 1 + 1.732i. Most calculators (this one included) return real-only results when possible and NaN for cases like (-1)^0.5. (3) For exponential growth/decay over time (compound interest, population, radioactive decay), use the dedicated compound-interest or related calculators, which apply this same math but with time-based formulas built in.
The formula
base is the number being multiplied. exp is the exponent (power). Both can be any real number — positive, negative, integer, fraction or decimal. The result is base × base × … (exp copies) for positive integer exp, and is generalised by the laws above for other cases.
Example calculation
- Compute 2¹⁰ (a common power-of-2 reference).
- 2¹⁰ = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 = 1024.
- Negative exponent: 2⁻³ = 1/2³ = 1/8 = 0.125. Fractional exponent: 27^(1/3) = ∛27 = 3.
Frequently asked questions
Why does anything to the 0 power equal 1?
It's the only definition consistent with the laws of exponents. Specifically, aᵐ / aᵐ = 1 (anything divided by itself), but also aᵐ / aᵐ = aᵐ⁻ᵐ = a⁰ by the subtraction rule. So a⁰ has to equal 1. The only exception is 0⁰, which is technically indeterminate — different contexts (combinatorics, calculus, programming) define it differently, often as 1 by convention.
How are exponents related to logarithms?
They're inverses. If bˣ = y, then logᵦ(y) = x. Exponentiation answers "what do you get?"; logarithms answer "what exponent do you need?". Concretely, log₁₀(1000) = 3 because 10³ = 1000. Logarithms are how you solve equations like "2ˣ = 50" — answer: x = log₂(50) ≈ 5.64. They're also how you measure growth rates that span many orders of magnitude (decibels, pH, earthquake magnitudes are all logarithmic). See the logarithm calculator for inverse-direction work.
Why does my calculator show "infinity" or "NaN" for some inputs?
"Infinity" appears when the result exceeds the maximum representable number (~10³⁰⁸ for IEEE 754 doubles). For example, 2^1024 overflows. "NaN" (Not a Number) appears for mathematically undefined cases: 0⁰, (-1)^0.5 (would need complex numbers), or 0^(-1) (1/0, divide by zero). For genuinely huge numbers, switch to symbolic computation (Wolfram Alpha) or arbitrary-precision libraries; the standard floating-point used in browser JavaScript can't represent them.