The Newton-Raphson method can be used to find approximations of the $n$-th root $x=\sqrt[n]{S}$ of a number $S$. For an approximation, we try to solve $$x^n−S=0$$ If $\widehat x$ is a guess to the value of $\sqrt[n]{S}$, then the following formula is always a better guess: $$ \widehat x - \frac {{\widehat x}^n - S}{n⋅\widehat{x}^{n-1}} $$ We can implement this recursively until the difference between consecutive guesses is smaller than some tolerance.