This paper is concerned with proving some embeddings of the form \begin{equation*}A_{p_{1},q_{1}}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })\cdot A_{p_{2},q_{2}}^{r}(\mathbb{R}^{n},|\cdot |^{\alpha })\hookrightarrowA_{p,q_{1}}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha }).\end{equation*} The different embeddings obtained here are under certain restrictions on theparameters. Here $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha})$ stands foreither the Besov space $B_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ orthe Triebel-Lizorkin space $F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$.These spaces unify and generalize the classical Lebesgue spaces of powerweights, Sobolev spaces of power weights, Besov spaces and Triebel-Lizorkinspaces. Almost all our assumptions on $s,r,p_{1},p_{2},q_{1}$ and $q_{2}$are necessary. An application to the continuity of pseudodifferentialoperators with non-regular symbols on Triebel-Lizorkin spaces of powerweights is given.