9000035810 | math4u.vsb.cz

SubArea:

Powers and roots of complex numbers

Level:

Project ID:

9000035810

Accepted:

1

Clonable:

0

Easy:

0

Given the complex number

z = - 2 + 2 i

, find all the roots of

\sqrt[3]{z}

.

\begin{aligned} w_{0} = \sqrt[6]{8} (\cos \frac{π}{4} + i \sin \frac{π}{4}) \\ w_{1} = \sqrt[6]{8} (\cos \frac{11 π}{12} + i \sin \frac{11 π}{12}) \\ w_{2} = \sqrt[6]{8} (\cos \frac{19 π}{12} + i \sin \frac{19 π}{12}) \end{aligned}

\begin{aligned} w_{0} = 2 (\cos \frac{π}{4} + i \sin \frac{π}{4}) \\ w_{1} = 2 (\cos \frac{11 π}{12} + i \sin \frac{11 π}{12}) \\ w_{2} = 2 (\cos \frac{19 π}{12} + i \sin \frac{19 π}{12}) \end{aligned}

\sqrt[3]{- 2} + \sqrt[3]{2}

\begin{aligned} w_{0} = 2 (\cos \frac{π}{3} + i \sin \frac{π}{3}) \\ w_{1} = 2 (\cos π + i \sin π) \\ w_{2} = 2 (\cos \frac{5 π}{3} + i \sin \frac{5 π}{3}) \end{aligned}