Question

Find all the 8th roots of i3 − 3. Also show any one of them in an Argand diagram.

Answer 1

Answer on Question #66711 – Math – Complex Analysis

Question

Find all the 8th roots of i3 – 3. Also show any one of them in an Argand diagram.

Solution

$$\begin{array}{l} z = -3 + 3i = 3\sqrt{2} \left( \cos \frac{3\pi}{4} + \mathrm{isin} \frac{3\pi}{4} \right) \\ \sqrt[8]{z} = \sqrt[8]{3\sqrt{2}} \left( \cos \frac{3\pi + 8\pi n}{32} + \mathrm{isin} \frac{3\pi + 8\pi n}{32} \right), \; n = 0, 1, \dots, 7. \end{array}$$ $$\begin{array}{l} z_1 = \sqrt[16]{18} \left( \cos \frac{3\pi}{32} + \mathrm{isin} \frac{3\pi}{32} \right); \quad z_2 = \sqrt[16]{18} \left( \cos \frac{11\pi}{32} + \mathrm{isin} \frac{11\pi}{32} \right); \\ z_3 = \sqrt[16]{18} \left( \cos \frac{19\pi}{32} + \mathrm{isin} \frac{19\pi}{32} \right); \quad z_4 = \sqrt[16]{18} \left( \cos \frac{27\pi}{32} + \mathrm{isin} \frac{27\pi}{32} \right); \\ z_5 = \sqrt[16]{18} \left( \cos \frac{35\pi}{32} + \mathrm{isin} \frac{35\pi}{32} \right); \quad z_6 = \sqrt[16]{18} \left( \cos \frac{43\pi}{32} + \mathrm{isin} \frac{43\pi}{32} \right); \\ z_7 = \sqrt[16]{18} \left( \cos \frac{51\pi}{32} + \mathrm{isin} \frac{51\pi}{32} \right); \quad z_8 = \sqrt[16]{18} \left( \cos \frac{59\pi}{32} + \mathrm{isin} \frac{59\pi}{32} \right); \end{array}$$

Argand diagram for $z_3 = \sqrt[16]{18} \left( \cos \frac{19\pi}{32} + \mathrm{isin} \frac{19\pi}{32} \right)$

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