Question

Find all the roots of 1+z^6=0, and show them in an argnad diagram

Answer 1

Given that $1 +Z^6 = 0$

$Z^6 = -1\\ Z = (-1)^{1/6}\\ Z = (-1+0*i)^{1/6}\\ Z_k = (cos\pi+ i*sin(2*k*\pi+\pi))^{1/6} \\ k = 0,1,2,3,4,5\\ Z_k = cos(2*k+1)*\pi/6$

• if k = 0:

$Z_0 = cos \pi/6+isin\pi/6 = \sqrt3/2+i*1/2$

• if k = 1:

$Z_1 = cos (3*\pi/6)+i*sin(3*\pi/6)= cos\pi/2+i*sin\pi/2 = 0+i*1$

• if k = 2:

$Z_2 = cos (5*\pi/6)+i*sin(5*\pi/6)= -\sqrt3/2+i*1/2$

• if k = 3:

$Z_3 = cos (7*\pi/6)+i*sin(7*\pi/6)= -\sqrt3/2+i*(-1/2)$

• if k = 4:

$Z_4 = cos (9*\pi/6)+i*sin(9*\pi/6)= cos(3*\pi/2)+ i*sin3*\pi/2=0+i*(-1)=0-i$

• if k = 5:

$Z_5 = cos (11*\pi/6)+i*sin(11*\pi/6)= cos(360-30)+ i*sin(360-30 )=\sqrt3/2-i*1/2$