  In this article, we give a combinatorial approach to the exponents of the
Moore spaces. Our result states that the projection of the $p^{r+1}$-th power
map of the loop space of the $(2n+1)$-dimensional mod $p^r$ Moore space to its
atomic piece containing the bottom cell $T^{2n+1}\{p^r\}$ is null homotopic for
$n>1$, $p>3$ and $r>1$. This result strengthens the classical result that
$\Omega T^{2n+1}\{p^r\}$ has an exponent $p^{r+1}$.
