Effective results on the waring problem for finite simple groups

Let G be a finite quasisimple group of Lie type. We show that there are regular semi simple elements x, y ∈ G, x of prime order, and |y| is divisible by at most two primes, such that (formula presented). In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of a previous result of Larsen, Shalev, and Tiep by showing that, given any integer m ≥ 1, if the order of a finite simple group S is at least (formula presented), then every element in S is a product of two mth powers. Furthermore, the verbal width of x^m on any finite simple group S is at most (formula presented). We also show that, given any two non-trivial words w1, w2, if G is a finite quasi simple group of large enough order, then (formula presented).