I think, in the case of an algebra class, the student could assume something like an integral / rational solution space. Otherwise, the solution is just to put the opposite symbol on the other side of the equation, e.g. (8192)^(1/3), which is maybe(?) what the teacher is asking, however would the teacher actually accept an answer written in the form 27^(1/3)? I would guess that I would not accept that. I understand what you are saying, but in this case, it could be argued that the 8192 case is actually a different question, because the underlying multiplicative group is different.

Why does it matter the complexity of the solution check?

If for some reason the counter example of the Reimann hyp would be checkable in exponential time (whatever that means), and if it would have required 10 years of computer time to check, how would that matter? The solution would still stand (assuming that everybody agrees that no mistake was made).

The question applies to the classroom setting we were talking about; teachers don't have access to supercomputers :P

To expand on my previous response: you want a proof that is checkable in polynomial time; yes - completely agree. I was just playing the devil's advocate case, that maybe integer solutions could be assumed (if it was a low-level class, for example).