Consider $\{B_i\}$ where $i\in I$ and $I$ is countable infinite. $|B_i|=|B_j|=n$ for all $i,j$ and $n \ge |\mathbb{N}|$. I want to show that $| \large \cup_{i\in I}$$B_i|=n$
I am given that $a*a=a$ for any infinite cardinal $a$ and I know that $|\mathbb{N}^k|=|\mathbb{N}|$
HINT: First show that in the case that the $B_i$ are pairwise disjoint, the union embeds into $n\times\Bbb N$. Then show that in the case it's not disjoint you can still embed it, with a slight twist. Then use what you are given.
Find an injection from $\bigcup B_i$ into $I\times B$, where $B$ is any set of cardinality $n$. Since $I$ is countable we can assume that $I=\Bbb N$.
Next find an injection from $B$ into $\Bbb N\times B$; and an injection from $\Bbb N\times B$ into $B\times B$. Now use the fact that $B\times B$ and $B$ have the same cardinality.