Language intersections (set theory) - Understanding better I've joined computer science classes at high school because I have a wide knowledge and a few years of experience in programming in multiple of languages, however I didn't fit in the requirements of the class, which is a specific mathematics level. I am a low level mathematics student, I don't really know formal mathematics or any proofs besides a simple proof like proving that an angle is 90 degrees.

So besides programming, there is this subject we started studying about finite deterministic automatons, stack automatons, turing machines and regular languages, etc.

I understand the concept very well, especially when it comes to building turing machines, finite and stack diagrams. At the class, we solve these automatons with diagrams, not in a formal way.

So besides that, my teacher gives us a lot of questions about union, intersection and I am getting really confused with intersection.

Surely I can understand the basic intersections with set of words:

$L = \{aab, bba, aa\}$
$L' = \{a, bbba, aab, aa\}$

So $L\cap L'$ would be $\{aab, aa\}$ 

I also understand basic language intersections like this:

$L = \{a^n b^n \mid n \ge 0\}$, $L' =  \{a^n \mid n \ge 0\}$

$L\cap L'$ would be $\{a^n \mid n \ge 0\}$

But when it comes to something like this:

$L = \{a^{2n} b^m a^k \mid n,m \ge 0, k = m \mod 2\}$

$L'$ = set of string elements such that the length $|w|$ is even

$L\cap L'$ = I don't know the answer for

The second part of the language which defines the length of the elements (e.g on $n$, of $m$, of $k$) really confuses me in intersections and unions, I am not really sure why. Does that part matters in intersections? I am always getting lost in these cases.

How can I understand this concept a bit better mathematically?
In your last example, when trying to deduce $L \cap L'$, the trick is to trying to understand which strings in $L$ have even length. You can do this as follows:

$$L = \{a^{2n} b^m a^k | n,m &gt;= 0, k = m \pmod 2\}.$$

We can forgot about the first substring of $a$'s, because they always come in pairs (or there are none. In either case, they contribute an even number of characters. Now consider $\{b^m a^k | m &gt;= 0, k = m \pmod 2\}$. The length of the strings in this set is $m+k$. Since $k = m \pmod 2$, we have $m+k = m + m = 2m = 0 \mod 2$. Therefore, $m+k$ is always even, and so $b^m a^k$ always contributes an even number of characters as well. Since $even + even = even$, every string in $L$ is even. Therefore, $L \cap L' = L$.