**Problem:** For which of the following complex numbers $z\in\mathbb{C}$ is $ z^8\in \mathbb{R}$?

a) $z=-5+5i$  

b) $z=100+i$

Of course, one can go the hard way and apply the binomial theorem or compute everything by hand, which will give us the following: 

a) $z^8 = 6250000\;$ and  

b) $z^8=9.972007 \cdot 10^{15} + 7.99440056 \cdot 10^{14} i$

**Question:** Is there a faster way to find out that $z\in\mathbb{R}$ for a) and $z\in\mathbb{C}\setminus\mathbb{R}$ for b)?