Exponent Calculator
Compute any base raised to any exponent. Supports negative bases, negative exponents and fractional exponents (which are the same as roots).
What is an exponent?
An exponent — sometimes called a power — tells you how many times a base multiplies itself. Two to the power of three, written 2³, is 2 × 2 × 2 = 8. Negative exponents flip the base into a reciprocal: 2⁻³ is the same as 1 / 2³ = 0.125. Fractional exponents are roots: 9^(1/2) is the square root of 9, which is 3. A negative base combined with a non-integer exponent (like (−4)^0.5) has no real value — this calculator flags that instead of returning silent NaN.
Exponents, one case at a time
Every exponent is a compact way of describing repeated multiplication, but the exact meaning shifts as the exponent becomes negative, fractional or zero. Each card below shows one case with the rule, a picture and a worked value.
Positive integer exponent — repeated multiplication
The most common case: multiply the base by itself as many times as the exponent tells you. The exponent is just shorthand for a chain of multiplications.
Negative exponent — flip into a reciprocal
A negative exponent means "one over the positive version". The base moves to the denominator and the exponent's sign flips to positive.
Fractional exponent — the same as a root
A fractional exponent takes a root of the base. The denominator of the fraction is the root index; the numerator is an ordinary power applied after (or before — either works).
Zero exponent — always 1 (for non-zero base)
Dividing a^m by a^m gives a^(m−m) = a^0, and any non-zero value divided by itself is 1. So a^0 = 1 for every non-zero base. 0^0 is left undefined in most algebra contexts.
Combining exponents — the three shortcut rules
When the same base appears in a product, quotient or nested power, you can combine the exponents without expanding them out. These are the rules you use most in algebra.
Features of this calculator
- Handles any real base with integer, decimal, negative, or fractional exponents
- Expands small integer powers into the underlying multiplication (e.g. 2³ = 2 × 2 × 2)
- Detects complex results (e.g. even root of a negative) and explains them clearly
- Works for very large and very small results without silent overflow
- Plots the growth curve y = a^x with your input point marked
Frequently asked questions
+What does a negative exponent mean?
A negative exponent is a reciprocal: a⁻ⁿ = 1 / aⁿ. It flips the base.
+Are exponents and roots the same?
Yes. A root is an exponent with a fractional value: ⁿ√a = a^(1/n).
+Why does 5⁰ equal 1?
Because a^m ÷ a^m = a^(m−m) = a^0, and any non-zero number divided by itself is 1.
+Is 0⁰ defined?
It is a special case. In most algebra contexts 0⁰ is left undefined; in combinatorics and many programming languages it is treated as 1 for convenience.