Updated · Built and checked by the PlinzoTools team, formulas tested against published sources
Calculate base raised to any exponent, including negative and fractional powers.
1024
This tool raises a base number to any exponent -- whole, negative, or fractional (which computes a root).
Enter the base and the exponent. The result appears instantly.
Result = baseexponent
A negative exponent gives the reciprocal (e.g. 2^-2 = 1/4); a fractional exponent gives a root (e.g. 8^(1/3) = 2, the cube root).
An exponent tells you how many times to use the base as a factor. In 3^4 the base 3 is multiplied by itself four times: 3 × 3 × 3 × 3 = 81. The tool goes beyond whole numbers, so you can also enter decimals, negative numbers, and fractions as the exponent. The result is displayed to about twelve significant digits, so very long decimals are trimmed and tiny floating-point noise is hidden.
5^3 = 5 × 5 × 5 = 125
10^-3 = 1 / 10^3 = 0.001
16^0.5 = the square root of 16 = 4
27^(1/3) = the cube root of 27 = 3
To calculate a fractional exponent such as 1/3, type the decimal value. Because 1/3 is 0.3333..., a decimal typed by hand is only an approximation, so the answer may differ from the exact root in the last digits. When the true value is a whole number you may see a result such as 2.99999999 rounded to 3 by the twelve-digit display, but for less tidy inputs treat the output as approximate.
You can use the calculator to verify any of these identities with your own numbers, which is a good way to build intuition.
A negative base works cleanly with whole-number exponents. The sign alternates: (-2)^3 = -8, while (-2)^4 = 16. Problems arise with fractional exponents. The square root of a negative number is not a real number, so an input like (-4)^0.5 gives no real result and the tool cannot display one. Some odd roots of negative numbers do exist as real values, such as the cube root of -8, but computing environments generally report them as undefined when the exponent is entered as a decimal, so do not rely on this. If you need the cube root of a negative number, take the cube root of its positive counterpart and attach the minus sign yourself.
Exponents grow numbers quickly. Doubling ten times gives 1,024, but doubling a hundred times gives a number with 31 digits. The tool uses standard double-precision arithmetic, so extremely large results may be shown in scientific notation, and results beyond the representable range cannot be displayed as a usable number. Similarly, very small results close to zero may lose precision. For exact big-integer arithmetic, a dedicated big-number tool is more appropriate.
It can produce a non-real result (e.g. the square root of a negative number) -- this tool returns "not a number" (NaN) in that case rather than a complex number.
Logarithms answer the inverse question -- "what exponent gives this result?" See the Logarithm Calculator.
Use a fractional exponent. The square root uses an exponent of 0.5 and the cube root uses roughly 0.3333333. For the cube root, the decimal is an approximation of one-third, so the last digits may be slightly off.
A negative exponent means reciprocal, not a negative result. 2^-3 = 1/8 = 0.125. The result is negative only when the base is negative and the exponent is an odd whole number.
Powers grow extremely fast, and the tool uses standard double-precision numbers. Past a certain size results cannot be displayed as an ordinary number, and only about twelve significant digits are meaningful.
Any nonzero base raised to the power 0 equals 1. This follows from the rule that dividing like bases subtracts exponents, so a^n divided by a^n gives a^0, which must equal 1.
Reference: NIST Weights and Measures