For even powers, the range of values being considered is important, and needs to be dealt with before doing any multiplication. For example, x^n for x \in [-1,1] should produce the interval [0,1] when n = 2, 4, 6, \ldots. But if [-1,1]^n is taken by repeating interval multiplication of form [-1,1] \cdot [-1,1] \cdot \cdots \cdot [-1,1] then the result is [-1,1], wider than necessary.
