Answer to Question #110806 in Complex Analysis for Nikesh

Question

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

Answer 1

$z=(-7)^{1/6}$

$z^6=-7$

$z^6=7e^{πi}$

$z^6=7e^{2πn+πi}$

$z=7^{1/6}e^{\frac{2πn+πi}{6}}$

$z_1=7^{1/6}e^{\frac{π}{6}i}$

$z_2=7^{1/6}e^{\frac{π}{2}i}$

$z_3=7^{1/6}e^{\frac{5π}{6}i}$

$z_4=7^{1/6}e^{\frac{7π}{6}i}$

$z_5=7^{1/6}e^{\frac{3π}{2}i}$

$z_6=7^{1/6}e^{\frac{11π}{6}i}$