I came across this problem in a complex analysis book:
Find the real and imaginary parts of $(1 + i)^{100}$.
Now, this question is asked before polar form is introduced, so I am curious about alternate methods of solution. Of course, I could calculate this brute force (break it down by factoring $100 = 2 \cdot 2 \cdot 5 \cdot 5$), but that is tedious. As far as I know, there aren't any multiplicative properties for $Re$ and $Im$ either.