I have extremly problems to calculate expressions with roots of unity.
How do I for example calculate something like this without the use of a calculator:
$$a=\frac{(e^{-i\frac{\pi}{4})^2}}{(e^{i\frac{\pi}{4}}-e^{-i\frac{\pi}{4}})(e^{i\frac{\pi}{4}}-e^{-3i\frac{\pi}{4}})(e^{i\frac{\pi}{4}}-e^{3i\frac{\pi}{4}})}$$
$$b=(e^{-3i\frac{\pi}{4}}+e^{-i\frac{\pi}{4}})$$
Thank you very much for your answer.
I think the best way to work with kind of expressions is to see the geometry. First of all, I would locate the points in the circle. Then I would transform these points into coordinates. For example, if $a=e^{-i\frac{\pi}{4}}$ and $b=e^{i\frac{\pi}{4}}$, then $$a=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$$ and $$b=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i.$$ Finally I would use the vector nature of these numbers: $$a^2=(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)=-i$$ and $$a-b=(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)-(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)=-\sqrt{2}i.$$ I hope this helps you. I apologize for my english, it's not my native language.