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Find all complex numbers $z$ such that $z^4 = -4.$

 

Note: All solutions should be expressed in the form , where  and  are real numbers.

 Feb 22, 2021
edited by Guest  Feb 22, 2021
 #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.

 Feb 22, 2021
 #2
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I don't understand it that well. Could you explain it? Thank you so much!!

Guest Feb 22, 2021

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