Find all the six, sixth roots of unity. Which of these are also cube roots of unity ?

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Solution

$(1)^{1 / 6} = cos \frac{2 m π}{6} + i sin \frac{2 m π}{6}$
where m = 0, 1,2,3,4,5
$= cos \frac{m π}{3} + i sin \frac{m π}{3}$
where m = 0, 1,2,3,4,5
$(1)^{1 / 3} = cos \frac{2 k π}{3} + i sin \frac{2 k π}{3}$
where k = 0, 1,2
The common values correspond to m = 0,2,4 or = 0,1,2 and they are
$1, cos \frac{2 π}{3} + i sin \frac{2 π}{3}, cos \frac{4 π}{3} + i sin \frac{4 π}{3}$
or 1, -1/2 + i $\sqrt{3}$ /2, -1/2 - i $\sqrt{3}$ /2

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