# How do we further simplify this expression involving a complex number?

I will state the problem here:

Let $$\omega$$ be a complex number such that $$\omega^5 = 1$$ and $$\omega \neq 1$$. Compute $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.$$

W is obviously a complex number. I've tried arranging and rearranging the terms in a way to further simplify the expression, but I'm having trouble getting anywhere with it. Anyone have ideas on how to proceed and solve the problem?

• Use the given condition to get $ω^4 = 1/ω$ and edit your question to include what you can get from that. Aug 5, 2020 at 18:28

The first and last terms are equal, as can be seen by multiplying the latter's numerator and denominator by $$\omega^2$$. A similar treatment, with a factor of $$\omega$$, shows the middle terms are equal too. So we want to double$$\frac{\omega}{1+\omega^2}+\frac{\omega^3}{1+\omega}=\frac{\omega+\omega^2+\omega^3+\omega^5}{(1+\omega)(1+\omega^2)}=1,$$giving $$2$$ as the final answer. We actually don't need to exclude $$\omega=1$$ for this to work.