Moments of products of independent random variables: $E[ X^kY^k ]$

Question

If we are given that two random variables $X$ and $Y$ are independent, I'm wondering if the rule: $E[XY] = E[X]E[Y]$ applies for any integer $k>0$, such that:

$E[X^kY^k] = E[X^k]E[Y^k]$.

Is this a straight forward result? or am I missing something fundamental?

Thanks for your comments / suggestions.

Answer 1

If $X$ and $Y$ are independent, so are $f(X)$ and $g(Y)$ where $f$ and $g$ are measurable.

Hence $X^k$ and $Y^k$ are independent. Hence, provided that they exist,

we have $$E[X^kY^k]=E[X^k]E[Y^k]$$