  We consider the curves $ y^2=x^{2^m} -c$ and $y^2=x^{2^{d}+1}-cx$ over the
rationals where $c \in \mathbb{Q}^{\times}.$ These curves are related via their
associated Jacobian varieties in that the Jacobians of the latter appear as
factors of the Jacobians of the former. One of the principle aims of this paper
is to fully describe their Sato-Tate groups and distributions by determining
generators of the component groups. In order to do this, we first prove the
nondegeneracy of the two families of Jacobian varieties via their Hodge groups.
We then use results relating Sato-Tate groups and twisted Lefschetz groups of
nondegenerate abelian varieties to determine the generators of the associated
Sato-Tate groups. The results of this paper add new examples to the literature
of families of nondegenerate Jacobian varieties and of noncyclic Sato-Tate
groups. Furthermore, we compute moment statistics associated to the Sato-Tate
groups which can be used to verify the equidistribution statement of the
generalized Sato-Tate conjecture by comparing them to moment statistics
obtained for the traces in the normalized $L$-polynomials of the curves.
