  We study a $(4+D)$-dimensional Kaluza-Klein cosmology with a Robertson-Walker
type metric having two scale factors $a$ and $R$, corresponding to
$D$-dimensional internal space and 4-dimensional universe, respectively. By
introducing an exotic matter in the form of perfect fluid with an special
equation of state, as the space-time part of the higher dimensional
energy-momentum tensor, a four dimensional effective decaying cosmological term
appears as $\lambda \sim R^{-m}$ with $0 \leq m\leq 2$, playing the role of an
evolving dark energy in the universe. By taking $m=2$, which has some
interesting implications in reconciling observations with inflationary models
and is consistent with quantum tunneling, the resulting Einstein's field
equations yield the exponential solutions for the scale factors $a$ and $R$.
These exponential behaviors may account for the dynamical compactification of
extra dimensions and the accelerating expansion of the 4-dimensional universe
in terms of Hubble parameter, $H$. The acceleration of the universe may be
explained by the negative pressure of the exotic matter. It is shown that the
rate of compactification of higher dimensions as well as expansion of
4-dimensional universe depends on the dimension, $D$. We then obtain the
corresponding Wheeler-DeWitt equation and find the general exact solutions in
$D$-dimensions. A good correspondence between the solutions of classical
Einstein's equations and the solutions of quantum Wheeler-DeWitt equation in
any dimension, $D$, is obtained based on Hartle's point of view concerning the
classical limits of quantum cosmology.
