Find all the sixth squareroot s of one in the set of complex

Find all the sixth squareroot s of one in the set of complex numbers.

Solution

z=1

z=e⁰ⁱ

sixth roots =z^1/6

z^1/6=e^{((2k +0)/6)i} ,k= 0,1,2,3,4,5

z^1/6=e^(k/3)i ,k= 0,1,2,3,4,5

z^1/6=e⁽⁰⁾ⁱ ,e^(/3)i ,e^(2/3)i ,e⁽⁾ⁱ ,e^(4/3)i ,e^(5/3)i

z^1/6=1,(1/2)+i(3 /2) ,-(1/2)+i(3 /2) ,-1 ,-(1/2)-i(3 /2) ,(1/2)-i(3 /2)

therefore 6th roots of one are 1,(1/2)+i(3 /2) ,-(1/2)+i(3 /2) ,-1 ,-(1/2)-i(3 /2) ,(1/2)-i(3 /2)