[stephencanon]

You have very clearly established the following:

    ab = ba --> (ab)^2 = a^2b^2

It is less obvious that you have proven that:

    (ab)^2 = a^2b^2 --> ab = ba

If I were grading a sophomore algebra class, I would expect to see something along the lines of:

    Suppose (ab)^2 = a^2b^2.
    Re-associating gives us a(ba)b = a(ab)b;
    multiplying on the left and right by the
    inverses of a and b gives the result.

In any domain outside of a sophomore algebra class, I happily accept much briefer and more hand-wavy proofs.

[dllthomas]

Ah, when I read your response I flipped the order of the problem around in recalling it, so had the if and only-if backwards. Yes, I was handwavy there but it seemed clear enough for the setting (which you seem to be granting anyway) - just wanted to be sure I wasn't misunderstanding something. Thanks :)