Hi, I'm having some problems trying to understand the problem where I have to find all the complex numbers z such thet z^4= -4, I need an Idea of how to solve this, please give me solution using normal alegebra.

icecreamlover Dec 19, 2020

#1**0 **

By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4). Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i. Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

Guest Dec 28, 2020

#2**0 **

Couldn't it also be sqrt(2i)?

Edit: Oh wait, never mind, because sqrt(2i) = sqrt((1+i)^2)

Pangolin14 Dec 28, 2020