An ideal I subset of k[x(1),..., x(n)] is said to have linear powers if I-k has a linear minimal free resolution, for all integers k > 0. In this paper, we study the Betti numbers of I-k, for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for d = 2,3 or n - 1, and the product of all ideals generated by s variables, for s = n - 1 or n - 2. We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.