Answer to Question #106340 in Complex Analysis for Nikesh

Question

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

Answer 1

$z=-7$

$x=Re(z)=-7$

$y=Im(z)=0$

$|z|=\sqrt{x^2+y^2}=7$

$\phi =arg(z)=\pi-arctan(y/|x|)=\pi-arctan(0/7)=\pi-0=\pi$

$z=7(cos\pi+isin\pi)$

To find the 6th roots we use this formula

$z_k=\sqrt[6]{z}=\sqrt[6]{|z|}(cos {\frac {\phi+2\pi k} 6}+isin{\frac {\phi+2\pi k} 6}), k=0,1,2,3,4,5$

So, let's find all $z_k$

$z_0=\sqrt[6]{7}(cos {\frac {\pi} 6}+isin{\frac {\pi} 6})$

$z_1=\sqrt[6]{7}(cos {\frac {\pi} 2}+isin{\frac {\pi} 2})$

$z_2=\sqrt[6]{7}(cos {\frac {5\pi} 6}+isin{\frac {5\pi} 6})$

$z_3=\sqrt[6]{7}(cos {\frac {7\pi} 6}+isin{\frac {7\pi} 6})$

$z_4=\sqrt[6]{7}(cos {\frac {3\pi} 2}+isin{\frac {3\pi} 2})$

$z_5=\sqrt[6]{7}(cos {\frac {11\pi} 6}+isin{\frac {11\pi} 6})$

Now, let's plot all roots on Argand diagram