Connection between zero covariance of two random variables and their conditional expectation

Question

I had an assignment to find covariance, conditional probability densities and conditional expectations of random variables $X$ and $Y$ with joint probability function $f(x,y) = 9x^2y^2$, $0 \le x \le 1$, $0 \le y \le 1 $. I found that $f(x) = 3x^2$, $f(y) = 3y^2$, $\mathbb{E}[X] = 3/4$, $\mathbb{E}[Y] = 3/4$, and $\mathbb{E}[XY] = 9/16$, so $\mathrm{cov}(X, Y) = 0$. Then I found conditional expectations, but my professor said that it is redundant because of zero covariance. Why it was unnecessary? The conditional expectations actually were identical to unconditional ones in my case.