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Find all complex numbers \(z\) such that \(z^4 = -4\).

 Jun 26, 2021
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the complex number -4 has a degree of 180 degrees. that means that the degrees of the solutions are going to be 180/4=45, 180/4+360/4=135, 180/4+360/4+360/4=225, and 180/4+360/4+360/4+360/4=315.

The magnitude of -4 is obviously 4, so the magnitude of the solutions are going to be 4^(1/4) = sqrt(2)

Therefore, the solutions in polar form are: \(\sqrt{2}(\cos(45)+i\sin(45)), \sqrt{2}(\cos(135)+i\sin(135)), \sqrt{2}(\cos(225)+i\sin(225)), \sqrt{2}(\cos(315)+i\sin(315))\)

Simplifying gives \(\boxed{-1-i, -1+i, 1-i, 1+i}\)

(notice that I gave a general way to solve this problem, but obviously, for this problem, you could factor it and solve it in a quicker way.)

 Jun 26, 2021

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