  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 as the space-time part of the higher dimensional
energy-momentum tensor, a 4-dimensional decaying cosmological term is appeared
as $\Lambda \sim R^{-2}$, playing the role of an evolving dark energy in the
universe. The resulting field equations yield the exponential solutions for the
scale factors. These exponential behaviors may account for the dynamical
compactification of extra dimensions and the accelerating expansion of the
4-dimensional universe in terms of the Hubble parameter. The acceleration of
universe may be explained by the negative pressure of the exotic matter. It is
shown that the rate of compactification of higher dimensions depends on the
dimension, $D$. We then obtain the Wheeler-DeWitt equation and find the general
exact solutions in $D$-dimensions. A good correspondence between the classical
solutions and the Wheeler-DeWitt solutions, in any dimension $D$, is obtained.
