Why is n^(1/m) no valid way to calculate a root

Question

So I came across a situation where a calculator only had square root, but I needed the cubic root. So I used the old $n^\frac13$ trick, and sure enough, the cubic root of n. So this got me thinking. As far as I have been tought there is no formula to calculate a root (only alorithms), but still $n^\frac1m = \sqrt[m]n$. Why is this not a way to calculate cubic roots without using an algorithm?

Answer 1

Calculating a power means calculating $\exp$ and $\ln$, as in $x^{\frac1n} = \exp(\frac1n \ln x)$. This is more expensive than for example using the Newton iteration to find the root.

You still need algorithms to compute $\exp$ and $\ln$.

Answer 2

That's brilliant! Let me use this calculate the cube root of 15!

Alright, no calculators, just a pen and paper.

$$15^{1/3}$$

Hmm.

...hmmm.

Oh.