  Let H_0 (resp. H_\infty denote the class of commuting pairs of subnormal
operators on Hilbert space (resp. subnormal pairs), and for an integer k>=1 let
H_k denote the class of k-hyponormal pairs in H_0. We study the hyponormality
and subnormality of powers of pairs in H_k. We first show that if (T_1,T_2) is
in H_1, then the pair (T_1^2,T_2) may fail to be in H_1. Conversely, we find a
pair (T_1,T_2) in H_0 such that (T_1^2,T_2) is in H_1 but (T_1,T_2) is not.
Next, we show that there exists a pair (T_1,T_2) in H_1 such that T_1^mT_2^n is
subnormal (all m,n >= 1), but (T_1,T_2) is not in H_\infty; this further
stretches the gap between the classes H_1 and H_\infty. Finally, we prove that
there exists a large class of 2-variable weighted shifts (T_1,T_2) (namely
those pairs in H_0 whose cores are of tensor form) for which the subnormality
of (T_1^2,T_2) and (T_1,T_2^2) does imply the subnormality of (T_1,T_2).
