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 encounter, from simple whole-number powers to negative and fractional ones.
| 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.
For a rigorous treatment of exponent rules, see Wolfram MathWorld on exponents and the Lamar University algebra notes on integer exponents.
Also on MathCalcTools: the derivative calculator (which applies the power rule to terms with exponents) and the integral calculator (which raises exponents by one).
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).
Editor at Encore Editorial, Chris Terry sets the editorial standards here and turns dense topics into plain English. He has written widely on education, finance, and consumer markets.