For arbitrary n, that won't have an inverse funtion that you can right down, even when such an inverse function exists.
I'll ignore all the special cases (a = 0, b = 0, solvability if n=2, what it means if n=1 or n = 0, etc.):
For there to even be an inverse, the function must be one to one on that domain, so you function must be strickly increasing or decreasing on .
Checking that, . Just by inspection, if a and b are both positive, or negative, then will be always positive or negative on .
If a and b have opposite sign, then find the real solutions for . Get .
Will have that , and , so it's either a local maxima or local minima - either way, y isn't one-to-one near .
Thus, when both a and b are non-zero, and n>2, yone to one on the domain of positive reals if and only if a and b have the same sign.
Note that the reason you can't write down the inverse, even when it has one, is that the practical procedure is to switch x & y, then solve for y, your inverse function.
Here that means: Solve for y: , or rewritten: . There's no general formula for that for all n.