Enter a base and an exponent to compute base^exponent. Supports negative and fractional exponents. The relevant law of exponents is shown below the result.
Try a negative exponent like -3, or a fractional one like 0.5 (which gives the square root).
An exponent tells you how many times to multiply the base by itself. A handful of rules cover every situation you will run into, from simple whole-number powers to negative and fractional ones that describe reciprocals and roots.
| Law | Formula | Example |
|---|---|---|
| Product rule | b^m * b^n = b^(m+n) | 2^3 * 2^4 = 2^7 = 128 |
| Quotient rule | b^m / b^n = b^(m-n) | 3^5 / 3^2 = 3^3 = 27 |
| Power of a power | (b^m)^n = b^(m*n) | (2^3)^2 = 2^6 = 64 |
| Zero exponent | b^0 = 1 (b not 0) | 7^0 = 1 |
| Negative exponent | b^(-n) = 1 / b^n | 2^(-3) = 1/8 = 0.125 |
| Fractional exponent | b^(1/n) = nth root of b | 8^(1/3) = 2 |
| General fractional | b^(m/n) = (nth root of b)^m | 8^(2/3) = 4 |
This is the default in the calculator. It uses repeated multiplication.
A negative fractional exponent combines two rules at once, which is where most confusion starts.
A negative exponent is not a negative number; b^(-n) flips the base into a fraction but the result stays positive whenever b is positive. Another common mix-up is applying the exponent to a negative base incorrectly: (-2)^2 = 4, but -2^2 = -4, because without parentheses the exponent binds to 2 first and the minus sign is applied after. With fractional exponents, remember the denominator picks the root and the numerator picks the power, in that order, not the reverse.
For a rigorous treatment of exponent rules, see Wolfram MathWorld's exponent page and the Lamar University algebra notes on integer exponents.
Exponents show up again the moment you start differentiating or integrating a polynomial.
An exponent tells you how many times to multiply the base by itself. In b^n, b is the base and n is the exponent. For example, 2^4 = 2 x 2 x 2 x 2 = 16.
A negative exponent means take the reciprocal. b^(-n) = 1 / b^n. For example, 2^(-3) = 1 / 2^3 = 1/8 = 0.125.
A fractional exponent represents a root. b^(1/n) is the nth root of b. So 8^(1/3) = the cube root of 8 = 2. More generally, b^(m/n) = the nth root of b raised to the m power.
Any nonzero number raised to the power of zero equals 1. This follows from the quotient rule: b^n / b^n = b^(n-n) = b^0, and any number divided by itself is 1.
The main laws are: product rule (b^m * b^n = b^(m+n)), quotient rule (b^m / b^n = b^(m-n)), power of a power ((b^m)^n = b^(m*n)), negative exponent (b^(-n) = 1/b^n), and zero exponent (b^0 = 1).
Marcus Vance spent a decade building data pipelines and testing calculation logic before he started writing about the math behind the tools. On MathCalcTools he checks every formula against the source math, then writes out the steps so a reader can follow the same path a calculator takes.