Surjective word maps and Burnside's p^a q^b theorem

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 (x,y)↦xNyN is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x,y,z)↦xNyNzN 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 (x,y)↦xNyN that depend on the number of prime factors of the integer N