Let G be a group generated by x and y, G 2 {G_2} be the commutator subgroup of G, and G 1 {G_1} be the group generated by y and G 2 {G_2} . This paper contains explicit expansions of y x m {y^{{x^m}}} modulo [ G 2 , G 2 , G 2 {G_2},{G_2},{G_2} ] and ( x y ) m {(xy)^m} modulo [ G 1 , G 1 , G 1 {G_1},{G_1},{G_1} ]. The motivation for these results stem from the p-groups of maximal class, for a large number of these groups have [ G 1 , G 1 , G 1 ] = 1 [{G_1},{G_1},{G_1}] = 1 .