[math-fun] Approximate solution to A^12 + B^12 = C^12

Somewhere in cyberspace I found a cartoon of Homer Simpson at a blackboard with some math and physics equations on it, and one of them is (and I quote): 3987^12 + 4365^12 = 4472^12 . This is false, but Mma says the 1/12th root of the LHS is 4472.000000007+. Question: How does one find such approximate equalities? Is it just a question of running through all numbers x of form x = (K^12 + L^12)^(1/12) for 1 <= K, L <= 9999, and picking the x nearest to an integer? Or is it cleverer than that? —Dan