Let $K$ be a field. Prove that $K(X,Y)$ is a finite extension of $K(X^2,Y^2)$ and find its degree.

Question

I conjectured that $K(X,Y)=K(X^2,Y^2)[X,Y]$ but to prove that I need to show $K(X,Y)\subset K(X^2,Y^2)[X,Y]$. I'm having problems showing any element in $K(X,Y)$ is of the form $X^mY^nF$ where $F\in K(X^2,Y^2)$ and $m,n$ nonnegative integers with no luck.