Do you expect three solutions, or just the real one? What about for x^2=-1, two or none? Is \sqrt[3]{7} enough, or is a decimal approximation required? If it isn't (and the student doesn't know how to go about it) one could argue that isn't much of an answer, but rather an existence claim, and that the methodology doesn't amount to much.
Personally I feel that you probably wouldn't give a question such as x^3=27 (neither would I), but if you did, marking it as wrong (as in no credit) after seeing the justification 1^3=1, 2^3=8, 3^3=27 would be too harsh. You can't penalize a student for giving out an easy question.
Cube root of 7 is the right answer. It doesn't have a finite decimal or repeating decimal answer so a decimal answer is wrong. But I wouldn't take off points for giving a decimal approximation as the answer.
In college algebra, for most sections, we deal with real number solutions. They haven't reached the point of knowing about non-real solutions. We teach at the level the students are at. Without having had trig finding the roots of unity is hard and not comprehensible to the students so asking them for all three solutions is a bit much.
I would not give a student in college algebra credit for solving x^3 = 27 by guess and check. It demonstrates that they really don't understand what is going on. I give credit for demonstrating understanding. Not demonstrating that they are good guessers.
Personally I feel that you probably wouldn't give a question such as x^3=27 (neither would I), but if you did, marking it as wrong (as in no credit) after seeing the justification 1^3=1, 2^3=8, 3^3=27 would be too harsh. You can't penalize a student for giving out an easy question.