Is the index in an nth root allowed to be fractional? We have nth roots that can be rewritten as fractional powers:
$$\sqrt[n] x = x^\frac 1 n$$
I was looking around on Wikipedia and some other online material, but I couldn't find any definitive set of numbers that the index, $n$, belongs to. 
Wikipedia says this in its nth root article:

The nth root of a number x, where n is a positive integer...

It then goes on to include this a few lines down:

For the extension of powers and roots to indices that are not positive integers, see exponentiation.

So one part states that $n$ must be positive integral, but another leads toward non-positive integral. However, I couldn't really pull anything from the Exponentiation article that was referenced.
Is it permitted for the index to be fractional, such that:
$$\sqrt[\frac 1 n] x = x^n$$

There's no mathematical reason for left-hand side of your equation to be invalid. It's just that the right-hand side is simpler and more conventional, just as one would prefer to write $a/b$ rather than $1/(b/a)$.
