<s>[INST] Complex numbers solving for all solutions 
*

*Find all complex numbers $z$ such that $z + \frac{1}{z}$ is real

*Let $a \gt b \gt 0$ be integers.  Find all the solutions in complex numbers to the equation $z^a = z^{-b}$
For the first one, I tried manipulating the equation $Re(z) = \frac{z + \bar{z}}{2}$ to equal $z + \frac{1}{z}$ but I'm not really getting anywhere.
Don't know where to start for the second.
 [/INST] Question 1

Let $z=x+iy$ for real numbers $x,y$. Clearly, they may not be both $0$ given we are dividing by $z$.  $$z + z^{-1} = (x+iy) + \left(\frac{x}{x^2+y^2} - \frac{iy}{x^2+y^2}\right).$$
  In order for $z+z^{-1}$ to be real, its imaginary part must be $0$. Therefore,
  $$y- \frac{y}{y^2+x^2}=\frac{y(x^2+y^2-1)}{x^2+y^2}=0.$$
  So, $y=0$ or $x^2+y^2=1$. That is to say, the imaginary part of $z$ is $0$ or $z$ lies on the unit circle. 

</s>