I disagree. Whether a process is general or not is not important, sometimes problems aren't part of an obvious larger class of problems, and even then using a method that only works for a small set of problems is not technically wrong, and can sometimes even be many times easier than using a fully general method. For example, solving x^3 = 27 by remembering 3^3 = 27 is a perfectly acceptable solution. Trial and error shouldn't so readily dismissed.

In my opinion 'show your work' means that you show why you've arrived at a particular answer, why this is a correct solution, and (if necessary) why this is the only solution.

Using that criterion, simply writing something like '3^3 = 27', or more properly 'Simple trial and error shows x=3 to be a solution, since 3^3 = 27', would suffice.

If the problem was instead x^3 = 7, then sure this method wouldn't work (although they might be able to figure out it's somewhere between 1 and 2) but then again 'By definition x=3√7.' isn't particularly illuminating either, even though it's a full and correct derivation of the answer.