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I'm really stuck on this question, and was hoping someone could help me.

1. The problem statement, all variables and given/known data

Find all complex numbers, z, that satisfy [tex]z^3 = \sqrt{3} - i[/tex]

2. Relevant equations

[tex]z^n = r^n cis (n\theta)[/tex]

[tex]r = \sqrt{a^2 + b^2}[/tex]

[tex]\theta = tan^{-1}(\frac{b}{a})[/tex]

Rotate anticlockwise by [tex]\frac{2\pi}{n}[/tex] radians

3. The attempt at a solution

[tex]z^3 = \sqrt{3} - i[/tex]

[tex]r = \sqrt{(\sqrt3)^2 + (-1)^2} = 2[/tex]

[tex]\theta = tan^{-1}(\frac{-1}{\sqrt3}) = -\frac{\pi}{6}[/tex]

[tex]z^3 = 2cis(-\frac{\pi}{6})[/tex]

This is where I go wonky.:shy:

[tex]n = \frac{1}{3}[/tex]

[tex](z^3)^{\frac{1}{3}} = 2^{\frac{1}{3}}cis(\frac{1}{3}.-\frac{\pi}{6})[/tex]

[tex]z = \sqrt[3]{2}cis(-\frac{\pi}{18})[/tex]

Now, from what I've seen in the lectures, I'm supposed to add [tex]\frac{2\pi}{\frac{1}{3}}[/tex] radians, then [tex]\frac{4\pi}{\frac{1}{3}}[/tex] radians, then [tex]\frac{6\pi}{\frac{1}{3}}[/tex] to get [tex]2\pi[/tex], and thus complete one revolution (360 degrees).

I've seen an Argand Diagram for this question divided into sections of 120 degrees, but I'm so confused.

Could anyone help me please? I'd very much appreciate it.

Thanks.

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# Homework Help: De Moivre's Theorem : z^3

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