How this calculator works
An nth root reverses raising to the nth power. An odd degree has one real root for every real radicand, including negatives. An even degree has a nonnegative principal root only when the radicand is nonnegative. Inverse degree mode uses logarithms and accepts an integer degree from 1 to 1000 only when signs and values agree. Zero and roots of magnitude one cannot identify a unique degree.
Formula
r = nth_root(x) means r^n = x. For positive x and r not equal to 1, n = ln(x)/ln(r). The real principal even root is nonnegative.
Worked example
The cube root of -125 is -5 because (-5)^3 = -125. The fourth root of 81 is 3. In reverse, a root of 3 and radicand of 81 determine degree 4.
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Frequently asked questions
Why is the square root of 9 not shown as plus or minus 3?
The radical symbol denotes the principal root, which is 3. Solving the equation x^2 = 9 has two solutions, a different task.
Can I enter a fractional degree?
No. This tool uses the usual positive integer index of a radical. Rational powers require additional domain conventions.
Does it return complex roots?
No. Negative radicands with an even degree are rejected because this page works in the real numbers.