So I've been wrestling with this for about an hour and am still not sure how to solve it.
The task is to find the Im part of the complex number (it has to be as short as possible, or else it'd be easy):
$$\frac{1}{(z^2 + zi)}$$
for $z=(\sqrt{2}-i)/3$, where $i = \sqrt{-1}$.
I've first tried to replace $z$ but it didn't turn out well as I get very big numbers and it's really hard to keep track of everything (it doesn't seem like the task I normally get, which solves easier, that's why I had decided to quit with that way.)
I've also tried multiplying the first expression with
$$\frac{z^2 - zi}{z^2 -zi}$$
This way I get the solution which is pretty small comparing to the first, includes big numbers, but seems like a good one. So I want to ask if there is another way of solving this and what is the correct solution and the way you got it.
I hope someone answers, as I really want to know how to solve this kind of expression.
EDIT: It has been solved in the first comment.
