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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.

 Dec 19, 2020
 #1
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 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.

 Dec 28, 2020
 #2
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Couldn't it also be sqrt(2i)?

 

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

 Dec 28, 2020
edited by Pangolin14  Dec 28, 2020

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