Answer to Question #106140 in Complex Analysis for khushi

Question

Obtain the 6th roots of (−7) , and represent them in an Argand diagram.

Answer 1

$z=\sqrt[6]{-7}\\ z^6=-7\\ z^6=7e^{\pi i}\\ z^6=7e^{(2\pi n+\pi) i} \quad\quad n=0,1,...5\\ z=7^{\frac{1}{6}} e^{\frac{2\pi n+\pi}{6} i} \quad\quad n=0,1,...5\\ z=1.383e^{(\frac{\pi n}{3}+\frac{\pi}{6}) i} \quad\quad n=0,1,...5$

six answers are,

$z_1=1.383e^{\frac{\pi}{6} i} \\ z_2=1.383e^{\frac{\pi}{2} i} =1.383i\\ z_3=1.383e^{\frac{5\pi}{6} i} \\ z_4=1.383e^{\frac{7\pi}{6} i} \\ z_5=1.383e^{\frac{3\pi}{2} i} =-1.383i\\ z_6=1.383e^{\frac{11\pi}{6} i}$