Power calculator

What raising to a power means

Raising to a power means multiplying a number by itself as many times as the exponent says: 2⁵ is 2×2×2×2×2, that is 32. It's the operation that shows up whenever something grows by a fixed percentage over and over: a sum earning 3% a year for ten years is multiplied by 1.03¹⁰, and a population doubling every period is multiplied by 2 raised to the number of periods.

Exponents that aren't whole numbers

A negative exponent doesn't give a negative result: it gives the reciprocal, one divided by the power. So 10⁻² is 1/100 = 0.01, which is why the metric prefixes are written that way (milli is 10⁻³, micro is 10⁻⁶).

A fractional exponent is a root: raising to 0.5 is the square root (9^0.5 = 3), raising to 1/3 is the cube root. And any number raised to zero is 1, zero itself included: 0⁰ = 1 is a convention, not a law of nature, but it's the one every calculator uses because it keeps the formulas consistent.

The two cases where the result isn't a number

A negative base with a decimal exponent has no result among the real numbers: (−8)^0.5 would mean the square root of a negative number, which doesn't exist. In that case a dash appears here, not a number and not the infinity sign either, which would mean something else.

And numbers have a ceiling: beyond about 1.8×10³⁰⁸ the computer has no way left to represent them and the result becomes infinity. You get there sooner than you'd think: 2 raised to 2000 is already past it. Well before the ceiling the result switches to scientific notation, which isn't an error: it's how numbers with three hundred digits are written.

Nearby tools

For the inverse operation, «what exponent do I need to get this number», there is Logarithm. For expressions with brackets and several operations the page is Scientific calculator, and for the multiples and submultiples of the metric system SI prefixes.