Efficient method to calculate a big n-th root of a number

Question

I need to implement an algorithm to calculate big n-th roots like $\sqrt[\leftroot{-2}\uproot{2}\mbox{13500}]{ 200}$. I have tried the Newton's method but it requires the calculation of very big numbers that lead to overflows even though I am using 256 bits. I noticed though that my scientific calculator can easily calculate roots like that and even bigger ones. What algorithm is it using?

Answer 1

You may want to use that $200^{1 \over 13500}=e^{\frac{\ln 200}{13500}}$. Then use Taylor expansion to calculate $e^x$.