Languages and their operations: determining equivalence
Nope. Let $A = \{a\}$, $B= \{b\}$. Then $A^* = \{\epsilon,a,aa,
\ldots\}$ and similarly for $B$. So, $A^*\cup B^*$ contains strings of repeated "a" or repeated "b". But $(A\cup B)^*$ contains strings like $ab$, which you can't get from $A^*$ or $B^*$.

I'd suggest looking at a book like Sipser for a good coverage of languages like these and how to prove properties like equivalence. In particular, he has a great number of exercises to work through. Good luck!