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Math Help - Solve for z

  1. #1
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    Solve for z

    Solve z^6 + 8 = 0 for z.

    I assume that there are "6 6th roots of unity"?
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  2. #2
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    Hello, Ideasman!

    Solve: . z^6 + 8 \:= \:0

    We have: . z^6 \;=\;-8  \;=\;8\bigg[\cos\left(\pi + 2\pi n\right) + i\sin\left(\pi + 2\pi n\right)\bigg]

    Hence: . z \;=\;(2^3)^{\frac{1}{6}}\bigg[\cos(\pi + 2\pi n) + i\sin(\pi + 2\pi n)\bigg]^{\frac{1}{6}}

    . . . . . . z \;=\;\sqrt{2}\bigg[\cos\left(\frac{\pi}{6} + \frac{\pi}{3}n\right) + i\sin\left(\frac{\pi}{6} + \frac{\pi}{3}n\right)\bigg]

    Then let n \:=\:0,1,2,3,4,5

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