Products of commutators in free groups

Let F be the free group of rank 2, generated by x and y, and let w(x, y) is an element of F' be a nontrivial word. We give elementary algebraic proofs and algorithms to (1) express [x, y](n) as a product of [n/2] + 1 commutators and show this is the best possible; (2) show that (w(x, y))(2) cannot be written as one w-word and if g not equal 1 is an element of w(F) then show that the minimal number of w-words required to express g(n) as their product tends to infinity with n. Other results for free groups of higher rank axe also presented.