For a potential of the type V(r) = -b(o)\r +SIGMA(k = 1)(n)b(k)r (k), where n = 6, 8, we obtain an exact analytic solution of the Schrodinger-like wave (k = 1) equation generalized to N dimensions. We make use of an ansatz for the eigenfunction and obtain a closed form expression for the energy eigenvalues at the cost of certain constraints on the potential parameters, b(k). A distinct feature for even power potentials is pointed out for the case b(o) = O.