  We prove surjectivity of certain word maps on finite non-abelian simple
groups. More precisely, we prove the following: if N is a product of two prime
powers, then the word map sending (x,y) to the product of the Nth powers of x
and y is surjective on every finite non-abelian simple group; If is an odd
integer, then the word map sending the triple (x,y,z) to the product of the Nth
powers is surjective on every finite quasisimple group. These generalize
classical theorems of Burnside and Feit-Thompson. We also prove asymptotic
results about the surjectivity of the word map sending (x,y) to the product of
the Nth powers of x and y that depend on the number of prime factors of the
integer N.
