Power and root calculator
Powers and roots within the supported real domain.
Fill in the fields and the result will appear here automatically.
Calculates aⁿ or a^(1/n) in the real domain. For example, 2¹⁰ = 1024 and the cube root of −8 is −2. A negative base requires an integer exponent in power mode and a positive odd integer index in root mode. For a nonnegative base, a positive fractional root index is also supported as a reciprocal-power operation, beyond the usual integer-index root.
How it works
Formula and logic
For a positive integer n, aⁿ is a product of n factors a. For nonzero a, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A rational exponent m/n combines an nth root and an mth power in its real domain. Root mode evaluates a^(1/n); for negative a it extracts the sign only when n is an odd integer. The fields accept numbers rather than expressions such as 1/3.
Example
Two to the tenth power is 1,024, and the cube root of 27 is 3.
Fields and units
- Operation — list option
- Number — unitless
- Exponent — unitless
How to use
- — Choose power or root.
- — Enter the base as a number.
- — Enter an exponent for power or a positive index for root; ∛27 uses index 3.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: positive integer-index roots and rational exponents
- Limitation
- Calculations and display are rounded. This page rejects 0⁰ as ambiguous input; that is a product convention. Zero is accepted only with a positive exponent or root index. With a negative base, integer exponents are limited in magnitude to 9007199254740991 and odd root indices to that positive limit. Overflow or loss of a nonzero result produces an error.
FAQ
Why can't I take the square root of a negative number?
Because any real number squared is non-negative, so no such real root exists. It does exist among complex numbers, but that is a different domain.
What about the cube root of a negative number?
That exists and is computed: ∛−8 = −2, since (−2)³ = −8. The same holds for any odd root.
What does a negative exponent mean?
One divided by the same power with a positive exponent: 2⁻³ is 1/2³, that is 0.125.
Why is any number to the power of zero equal to one?
For nonzero a, aⁿ/aⁿ = a⁰ gives one. This page rejects 0⁰ rather than choosing one of its conventions for the user.
Can I use a fractional exponent?
Yes, for a nonnegative base: exponent 0.5 gives the square root. The field does not parse 1/3; use root mode with index 3 for a cube root. Fractional powers of negative bases are outside this calculator, even though some rational cases exist mathematically.