If you're just supposed to expand, then start by grouping together terms of the same base.
[tex](a^xb^yc^z)(a^rb^sc^t)=(a^xa^r)(b^yb^s)(c^zc^t)[/tex]
which is true since multiplication is commutative/associative. By definition of exponentiation, we have
[tex]a^xa^r=a^{x+r}[/tex]
and the same goes for the remaining bases. So the fully simplified form of the expression above would be
[tex]a^{x+r}b^{y+s}c^{z+t}[/tex]
