Is the Kleene star of an intersection always equal to the intersection of kleene stars?

Question

I know that the Kleene star of an intersection is contained in the intersection of Kleene stars, but are they necessarily equal?

For example, given two formal languages, $A$ and $B$, I know that $(A\cap B)^*\subseteq A^*\cap B^*$, but does $(A\cap B)^*= A^*\cap B^*$?

Answer 1

Clearly not.

Let $A=\{a\}$ and $B=\{aa\}$.

Now,

$A\cap B = \emptyset$

so

$(A\cap B)^* = \{\epsilon\}$

but

$A^*\cap B^*=B^*=\{a^{2i} : i \in \mathbb{N}\}$

(all strings consisting of an even number of $a$).