Q:

Real and Imaginary Parts of $(1 + i)^{100}$

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.

A:

An alternate method would be using the fact that $(1+i)^2 = 2i$. So then we have $(1+i)^{100} = (2i)^{50} = 2^{50}\cdot (i)^{50} = -2^{50}$.
For this case, the fact that $(1+i)^2=2i$ made this very easy to break down.

