Exponent Calculator
Calculate base raised to an exponent (base^exponent), including negative and fractional exponents, with clear domain rules for zero and negative bases.
How It's Calculated
Formula
\text{result} = \text{base}^{\text{exponent}}Exponentiation raises a base number to a power: base^exponent means the base multiplied by itself exponent times (for positive whole-number exponents), extended to negative and fractional exponents by standard rules. A negative exponent means take the reciprocal of the positive-exponent result (2^-3 = 1 / 2^3 = 0.125). A zero exponent always gives 1 for any nonzero base (5^0 = 1). A fractional exponent like 0.5 represents a root — 4^0.5 is the square root of 4, which is 2 — and this only has a real-number answer when the base is zero or positive; a negative base raised to a non-integer exponent (like (-8)^0.5) has no real-valued result, so this calculator rejects that combination rather than returning an invalid number. Zero raised to a negative exponent is also rejected, since it's equivalent to dividing by zero. Zero raised to the zero power (0^0) is a special case with no single universally agreed answer in mathematics; this calculator follows the common programming and combinatorial convention and returns 1.
Worked Examples
Positive integer exponent: 2^3
- 2^3 = 2 x 2 x 2
- Result: 8
Negative integer exponent: 2^-3
- A negative exponent means take the reciprocal: 2^-3 = 1 / 2^3
- 2^3 = 8, so 2^-3 = 1 / 8
- Result: 0.125
Frequently Asked Questions
Why does any number raised to the power of 0 equal 1?
This follows from the pattern of dividing consecutive powers: 2^3 / 2^2 = 2^1 = 2, and 2^1 / 2^1 = 2^0. Since any nonzero number divided by itself is 1, the pattern requires x^0 = 1 for every nonzero x, which is why this is a universal mathematical convention.
What does a negative exponent mean?
A negative exponent means 'take the reciprocal of the positive-exponent result.' base^-n is defined as 1 / base^n. For example, 2^-3 = 1 / 2^3 = 1/8 = 0.125.
Why is a negative base with a fractional exponent rejected?
A fractional exponent represents a root (0.5 is a square root, for instance). The square root of a negative number is not a real number — it's only defined in the complex number system, which this calculator doesn't support. So combinations like (-8)^0.5 are rejected as having no real-valued result, rather than silently returning an invalid value.
What is 0 raised to the power of 0?
Mathematicians don't universally agree on a single answer for 0^0 — it's treated as indeterminate in some contexts (like calculus limits) and as exactly 1 in others (like combinatorics and most programming languages). This calculator follows the common combinatorial/programming convention and returns 1, documented here as a deliberate policy choice rather than an accident.