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 = b

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.