You and I think that the cube root function is the simplest approach. The student doesn't necessarily agree. And I disagree with you that the most general approach is necessarily the simplest.
> Just about any method is valid in that type of problem.
Why is that not true for other types of problems?
> Passing a class should mean more than I got a lot of answers correct.
I agree. But you shouldn't penalise the student if the exam question is poorly framed (and we all make such mistakes). Just take a note for later and don't make the mistake again.
You have to be very cautious when marking wrong an answer that is correct, because you're at risk of confusing the student. The guess-and-try approach used in this particular example is very brittle, but it is correct. Professional mathematicians use it frequently, especially with the advent of computer-aided mathematics, for instance to find counter-examples to statements.
I see at least three issues with claiming that answer as wrong: first, correctness is essential (in the true sense) in mathematics, and therefore should not be carelessly dismissed in front of the student. Second, students should not be made to believe that guess-and-try is always inappropriate, but rather to understand that it won't always work. Finally, in this particular example the approach chosen is arguably (at least from the student's perspective) simpler than the one expected by the professor. Invalidating a "simpler" approach might give the student the impression that you always need to take the complicated route (ie, "math is hard") when the opposite is true.
My own take on this example would be to give (partial?) marks, with a lengthy comment of the type "fair enough, in this case, but what about if you wanted to solve x^3 =7? Your method wouldn't work, then!". Alternatively, if you don't want to give marks, it should be justified at length by rules clearly explained before the exam, while acknowledging the correctness of the approach.
> Just about any method is valid in that type of problem.
Why is that not true for other types of problems?
> Passing a class should mean more than I got a lot of answers correct.
I agree. But you shouldn't penalise the student if the exam question is poorly framed (and we all make such mistakes). Just take a note for later and don't make the mistake again.