i) Find all complex solutions to the equation z^4 +1 -i*3^(1/2) = 0
I basically have no clue, any tips/advice/solutions would be great.
I could also need some help with another question, this one is for the "experts" out there, (marked as more difficult in the text book)
ii) Show that the sum of the n nth roots of unity is zero. Hint: Show that these roots are all powers of the principal root.
I will be more than enough satisfied with just some help on the first one :) thanks in advance tho:DD
The equation is $$z^4=-1+\sqrt{3} i=2(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)$$ which is the same as $$z^4=2(\cos\frac{2\pi}{3}+i\sin \frac{2\pi}{3})$$
So $$z=\sqrt[4]{2}(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6})$$ and adding multiples of $\frac{\pi}{2}$, $$z=\sqrt[4]{2}(\cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3})$$ $$z=\sqrt[4]{2}(\cos \frac{7\pi}{6}+i\sin \frac{7\pi}{6})$$ $$z=\sqrt[4]{2}(\cos \frac{5\pi}{3}+i\sin \frac{5\pi}{3})$$