$A^k =$ {$(a_1, a_2, ... , a_k): a_1, a_2, ... , a_k \in A$}
and
$A^k =$ {$(a_1, a_2, ... , a_k) \in A$}
Is the second one just a short notation for saying the set of all tuples of length "k" that are made up of elements in A with replacement, but order matters?
Thanks!
No. Your second line picks the elements of $A$ that are ordered lists of length $k$. If $A$ contains many elements, but none that are ordered lists of length $k$, then the set you have written is empty (and it should not be).