Fastest nth root algorithm to a lot of digits?
An iterative method may be as follows:

$x_2=\frac{x_1(n-1)+\frac{x}{x_1^{n-1}}}{n}$

where $n$ is the root desired, $x_1$ is a guess of the root of $x$  and $x_2$ is a better guess.

The iterative method is given in the book The Art of Programming Embedded Systems by Jack G. Ganssle.