Level:
Project ID:
9000035810
Accepted:
1
Clonable:
0
Easy:
0
Given the complex number \(z = -2 + 2\mathrm{i}\),
find all the roots of \(\root{3}\of{z}\).
\(\begin{aligned}[t] &w_{0} = \root{6}\of{8}\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right ) &
\\&w_{1} = \root{6}\of{8}\left (\cos \frac{11\pi }
{12} + \mathrm{i}\sin \frac{11\pi }
{12}\right )
\\&w_{2} = \root{6}\of{8}\left (\cos \frac{19\pi }
{12} + \mathrm{i}\sin \frac{19\pi }
{12}\right )
\\ \end{aligned}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right ) &
\\&w_{1} = 2\left (\cos \frac{11\pi }
{12} + \mathrm{i}\sin \frac{11\pi }
{12}\right )
\\&w_{2} = 2\left (\cos \frac{19\pi }
{12} + \mathrm{i}\sin \frac{19\pi }
{12}\right )
\\ \end{aligned}\)
\(\root{3}\of{-2} + \root{3}\of{2}\)
\(\begin{aligned}[t] &w_{0} = 2\left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right )&
\\&w_{1} = 2\left (\cos \pi +\mathrm{i}\sin \pi \right )
\\&w_{2} = 2\left (\cos \frac{5\pi }
{3} + \mathrm{i}\sin \frac{5\pi }
{3}\right )
\\ \end{aligned}\)