Formula for the $n^{th}$ root of $n$?

Question

Is there a general formula $r(n)$ to calculate the nth root of $n$, for any complex, irrational, real, etc. $n$?

Sorry for formatting, I’m on mobile.

Answer 1

There is nothing specific to $\sqrt[n]n$ compared to $\sqrt[n]m$.

A general formula is

$$\sqrt[n]m=e^{(\log m)/n}.$$

Answer 2

The big difference is between

• natural, integer, rational, real numbers: for $n>0$ we have that $m=\sqrt[n] n$ is the unique value such that $m^n=n$
• complex number: the solution is not unique, for example for $n>0$ integer $m=\sqrt[n] n$ has $n$ distinct solutions

Answer 3

You can use Newtons Method to find nth roots recursively to an arbitrary precision.