Question 1
Let $\mathbb{Q}_{\gt 0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}_{\gt 0} \rightarrow \mathbb{Q}_{\gt 0}$ satisfying
$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
$$
for all $x, y \in \mathbb{Q}_{\gt 0}$.
$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
$$
for all $x, y \in \mathbb{Q}_{\gt 0}$.