Exponent Calculator

Raise a number to a power, marked exact only when the result is provably exact.

Worked example

Inputs: base 2, exponent −3. Formula: the result is the base raised to the exponent. Substitution: 2 to the power of −3 is 1 divided by 2 to the power of 3, which is 1 ÷ 8. Result: 0.125.

How it works

A positive whole-number exponent means repeated multiplication, a negative exponent means the reciprocal, and a fractional exponent means a root: 9 to the power of 0.5 is the square root of 9. Everything stays in the real numbers. For a negative base this calculator supports whole-number exponents only, so (−2) to the power of 3 is calculated but a fractional exponent such as 0.5 is not; this is a limit of the calculator, not a claim that no real value exists, because some fractional powers of a negative number, such as the cube root of −8, are real. Zero to a negative power is rejected because it divides by zero. This calculator defines 0⁰ as 1, following a common computational convention; in some mathematical contexts 0⁰ is left undefined or treated as indeterminate. An exact result is shown as a plain number and a rounded one with a leading "≈" (10 significant figures); the working line writes it as an equation with "=" or "≈". Results too large or too small to represent are reported rather than shown as infinity or zero.

What this result means

The result is the base raised to the exponent. A plain number means the digits shown are the exact value; a leading "≈" means the result is rounded, here to 10 significant figures.

Common mistakes

Reading -2² as (-2)². This calculator raises the number you enter, sign included, to the power: a base of -2 and an exponent of 2 gives 4. To get the negative of 2², calculate 2² and change the sign.
Treating 0⁰ as settled. This calculator defines 0⁰ as 1, following a common computational convention; in some mathematical contexts 0⁰ is left undefined or treated as indeterminate.

Assumptions and limitations

Real numbers only. For a negative base, only whole-number exponents are supported: (-2) to the power 3 is calculated, but a fractional exponent such as 0.5 is declined. This is a limit of the calculator, not a claim that no real value exists, since some fractional powers of a negative number, such as the cube root of -8, are real.
0 to a negative power is undefined (it divides by zero) and is rejected.
Results larger than about 1.8 × 10³⁰⁸ or smaller in size than about 5 × 10⁻³²⁴ cannot be represented and are reported instead of shown.
A result is marked exact only when the base and exponent have up to 15 significant digits and the power is provably a finite decimal.