Formula for the $n^{th}$ root of $n$? [closed]
By Emma Payne •
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.
$\endgroup$ 23 Answers
$\begingroup$There is nothing specific to $\sqrt[n]n$ compared to $\sqrt[n]m$.
A general formula is
$$\sqrt[n]m=e^{(\log m)/n}.$$
$\endgroup$ 3 $\begingroup$You can use Newtons Method to find nth roots recursively to an arbitrary precision.
$\endgroup$ 1 $\begingroup$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
