Find $ E[(1-X)^k]$ given $ E[X^k] = \frac{1}{(k+1)^2}$

The previous part of the question involved finding $ E[X^k]$ given $ X=AB$ , where $ A$ and $ B$ are uniformly distributed over $ [0, 1]$ . However, I don’t think this helps with the matter at hand.

My intuition was to expand $ E[(1-X)^k]$ as a series, which gave me a nasty sum I couldn’t compress. Is there a more direct approach? Can I avoid using $ A$ and $ B$ ?