Answer
The product of perfect squares is always a perfect square
The quotient of perfect squares is not always a perfect square
Work Step by Step
Let $a^{2}$ and $b^{2}$ be the squares of integers $a$ and $b$. The product of the squares equals
$$a^{2}b^{2}=(a\cdot a)(b\cdot b) $$
Since multiplication is associative, we can select any two neighboring factors to be multiplied first
$$ a^{2}b^{2}=a\cdot(ab)\cdot b $$
Since multiplication is commutative, the first two factors can change places:
$$\begin{align*}
a^{2}b^{2}&=(ab)\cdot a\cdot b & & \\
& =(ab)(ab) \\
& =(ab)^{2} \end{align*}$$
so the product of perfect squares is a perfect square, because $ab$ is a product of integers (an integer.)
Now, let $a=5$ and $b=2$. The quotient of $a^{2}=25$ and $b^{2}=4$ is
$$ \frac{a^{2}}{b^{2}}=\frac{25}{4},$$
which is not an integer.
