SOLUTION
We want to find the cube root of
[tex]4-4i\sqrt{3}[/tex]
This can be written as
[tex]\sqrt[3]{(4-i.4.\sqrt{3})}[/tex]
Step 2
Step 3
Step 4
Step 5
So, 3 degree root from input complex number has exactly 3 values:
These become
[tex]\begin{gathered} \alpha_0=2.(cos(-\frac{\pi}{9})+i.sin(-\frac{\pi}{9})) \\ \alpha_1=2.(cos(-\frac{5.\pi}{9})+i.sin(\frac{5.\pi}{9})) \\ \alpha_2=2.(cos(-\frac{11.\pi}{9})+i.sin(\frac{11.\pi}{9})) \end{gathered}[/tex]
Hence the answer is
[tex]\begin{gathered} \alpha_0=2.(cos(-\frac{\pi}{9})+i.sin(-\frac{\pi}{9})) \\ \alpha_1=2.(cos(-\frac{5.\pi}{9})+i.sin(\frac{5.\pi}{9})) \\ \alpha_2=2.(cos(-\frac{11.\pi}{9})+i.sin(\frac{11.\pi}{9})) \end{gathered}[/tex]
