8. Let $F$ be the set of all sequences of the form $\left(A_{1}, A_{2}, \cdots, A_{n}\right)$, where $A_{i}$ is a subset of $B=\{1,2, \cdots, 10\}$, and let $|A|$ denote the number of elements in the set $A$. Then $\sum_{F}\left|\bigcup_{i=1}^{n} A_{i}\right|=$ $\qquad$
8. $10\left(2^{10 n}-2^{9 n}\right)$.

Just calculate the number of times elements appear in set $B$. If $x \notin \bigcup_{i=1}^{n} A_{i}$, then $x$ does not belong to each of $A_{1}, A_{2}, \cdots, A_{n}$; if $x \in \bigcup_{i=1}^{n} A_{i}$, then $x$ belongs to at least one of $A_{1}, A_{2}, \cdots, A_{n}$. Therefore, to calculate the number of times $x$ appears, just subtract the number of $\bigcup_{i=1}^{n} A_{i}$ that do not contain $x$ from the total number of $\bigcup_{i=1}^{n} A_{i}$. Hence, the result is $10\left(2^{10 n}-2^{9 n}\right)$.