Is the Kleene star of an intersection always equal to the intersection of kleene stars? 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 $(A\cap B)^*= A^*\cap B^*$?
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 $a$).