Relation on $\mathbb N$:
$x \sim y \iff xy$ is a square
Give a description of the equivalence classes $[3]$, $[9]$, and $[99]$.
I'm not sure what this question is really asking, but this is what I have for $[3]$:
$\{y\in \mathbb N\ |\ 3y \text{ is a square }\}$
Any $y$ of the form $3$ times a square. Is this sufficient?
You can be a little more specific. Ideally, you could give a formula that generates the entire equivalence class.
Using the work you've already done, you know that in order for $3y$ to be a square, all of its prime factors need to be present with even multiplicity. Thus, $y$ needs to contain another factor of $3$ and takes on the form $$ y = 3z$$ for some $z \in \mathbb{N}$.
Next, for $3(3z) = 9z$ to be a perfect square, $z$ itself must be a perfect square. Thus, $y$ takes on the form $y = 3n^2$ for some $n \in \mathbb{N}$. Altogether, $$ [3] = \{3n^2 \mid n \in \mathbb{N}\}$$
In an analogous fashion, you can compute $[9]$ and $[99]$ by doing a prime-factorization on an equivalence class representative.