Question

Find all complex numbers such that:

z^4 = -1

How do I do this?

Do I take the forth root of -1?

Answer 1

take the forth root of -1 or write -1 as \(i^2\)\[z^4-i^2=0\]\[(z^2-i)(z^2+i)=0\]\[z=\pm\sqrt{i}\]\[z=\pm\sqrt{-i}\]

Answer 2

second is\[z=\pm i\sqrt{i}\]actually

Answer 3

and finally use the fact that\[i=\exp(\frac{\pi}{2} i)\]

Answer 4

makes sense?

Answer 5

I think so I just find the roots, that shouldn't be too hard I just need to consult my notes from previous class so,

z^(4) = -1
I would just write as

z = -1^(1/4) and then just find all the roots by taking 2pi intervals of the first root I obtain, and check to see if the solutions are unique

Answer 6

right :)