Prove that class of regular languages is closed on operation

Question

Let's have an operation $$\odot(L)=\{w\in L \; | \; |w|=2k \land k>0\}$$Show that result of this operation will be regular.

PS: It's not homework, it's from last year's exam.

Answer 1

The intersection of two regular languages is regular.

Let $E = \{ w : |w| \text{ even}, |w| > 0\}$. It's easy to show that $E$ is regular. It follows that $\odot(L) = E \cap L$ is regular too.