The Weyl law estimates the distribution of eigenvalues of the Laplace-Beltrami operator on a compact manifold. It states the number of eigenvalues less than $\lambda$ is $C\lambda^d + O(\lambda^{d-1})$, where $d$ is the dimension of the manifold. The remainder can be improved to a little-$o$ if the manifold does not have many closed geodesics, and can be improved by a logarithm with some stronger assumptions (e.g. nonpositive curvature). Situations for which the remainder improves by a power of $\lambda$ seem elusive. Here, we explore the prospects of improving the remainder term by a power for Cartesian products of manifolds. In particular, we improve the remainder by a power for products of spheres, and we investigate the relevant geodesic dynamics for general products.