  An ideal $I \subset \mathbb{k}[x_1, \ldots, x_n]$ is said to have linear
powers if $I^k$ has a linear minimal free resolution, for all $k$. In this
paper we study the Betti numbers of $I^k$, for ideals $I$ with linear powers.
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.
Especially, we are interested in ideals for which the Rees ideal is generated
by quadratic elements. This is related to a conjecture on matroids by White.
