Enter a base and an exponent to get the result, with a clear warning whenever the answer isn't a real number.
A negative exponent means "take the reciprocal": x^−n = 1/x^n. So 2^−3 = 1/2^3 = 1/8 = 0.125. The bigger the (positive) magnitude of a negative exponent, the closer the result gets to zero, since you're dividing by an ever-larger power.
A fractional exponent represents a root: x^(1/n) is the nth root of x. x^(1/2) is the square root, x^(1/3) is the cube root, and so on. A fraction like x^(2/3) combines the two ideas — take the cube root of x, then square the result (or square first, then take the cube root; both give the same answer for positive x).
It often has no real-number answer. Since a fractional exponent implies a root, and even roots (square root, 4th root, etc.) of a negative number aren't real — there's no real number that squares to a negative value — the math produces a complex result instead. This calculator detects that case and shows a warning rather than a misleading number. Odd roots of a negative base, like a cube root, do have a real answer (see the Root Calculator for that case handled directly).
Any nonzero number raised to the power of 0 equals 1. This follows directly from the exponent division rule: x^n ÷ x^n = x^(n−n) = x^0, and anything divided by itself is 1. It holds true regardless of how large or small the base is, as long as it isn't zero — 0^0 is a special edge case that different areas of math define differently depending on context.
Worked example: base 2, exponent 10: 2^10 = 2×2×2×2×2×2×2×2×2×2 = 1,024. Switching the exponent to −10 instead gives 2^−10 = 1/1,024 ≈ 0.000977, and an exponent of 0.5 gives 2^0.5 ≈ 1.41421, the square root of 2.
Exponents are repeated multiplication, and every rule follows from that. 2¹⁰ = 1024 because it is 2 multiplied by itself ten times.
Those last two trip people up most. 2⁻³ is 0.125, not −8.
2¹⁰ = 1,024. 2²⁰ = 1,048,576. 2³⁰ is over a billion. Doubling ten more times multiplies by a thousand, and this is the intuition that fails almost universally — people reason about exponential processes linearly and are consistently astonished by the result.
It is also why compound interest works, why epidemics accelerate, and why a storage figure that seemed generous becomes inadequate so suddenly. The Compound Interest Calculator is the same arithmetic applied to money.
Exponents bind tighter than multiplication and negation. −3² is −9, because the exponent applies to 3 before the minus sign; (−3)² is 9. Stacked exponents evaluate right to left, so 2^3^2 is 2⁹ = 512, not 8² = 64.
These are the two most common sources of a wrong answer when transcribing an expression into a calculator. If in doubt, add brackets — they cost nothing and remove the ambiguity entirely.
For the inverse, see the Log Calculator.
Very large powers lose digits in ordinary floating point — the Big Number Calculator keeps every one, and the Scientific Notation Calculator handles the magnitudes.