We study a (4 + D)-dimensional Kaluza–Klein cosmology with a Robertson–Walker type metric having two scale factors a and R, corresponding to a D-dimensional internal space and four-dimensional universe, respectively. By introducing exotic matter as the spacetime part of the higher dimensional energy–momentum tensor, a four-dimensional decaying cosmological term appears as Λ ~ 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 behaviours may account for the dynamical compactification of extra dimensions and the accelerating expansion of the four-dimensional universe in terms of the Hubble parameter. 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 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.