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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?

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    $\begingroup$ Use the given condition to get $ω^4 = 1/ω$ and edit your question to include what you can get from that. $\endgroup$
    – user21820
    Aug 5 '20 at 18:28
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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.

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  • $\begingroup$ I really appreciate your answer. Before I posted here, I managed to find that the first and last terms were equal, and the middle two terms were equal as well. I wasn't sure how to move on from there. $\endgroup$
    – mathtase
    Aug 5 '20 at 19:10

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