  The product $G\times H$ of two graphs is {\em trivially colorable} iff any
$\chi(G\times H)$-coloring of $G\times H$ is induced by a coloring of a factor.
A finite graph $G$ is said to be {\em tight} if any two $\chi(G)$-colorings of
$G$ differ in at least two points. Greenwell and Lov\'asz proved that if $G$ is
connected and tight with $\chi(G)\ge 3$, then $G\times K_{\chi(G)}$ is
trivially colorable. Strengthening their result, we prove that $G\times H$ is
trivially colorable provided $\chi(G)=\chi(H)\ge 3$, $G$ is connected and
tight, and $H$ is {\em weakly cliqued}, i.e. every vertex of $H$ is in a clique
of size $\chi(H)$. Using this theorem we show that every connected weakly
cliqued graph with $\chi(G)\ge 3$ is {\em trivially power-colorable}, i.e.
every finite power of $G^n$ is trivially colorable. We also characterize the
trivially power-colorable cographs. Finally, we describe the colorings of
infinite powers of trivially power-colorable graphs.
