How do you use DeMoivre's theorem to simplify (-1/2-sqrt3/2i)^3?

1 Answer
George C.
Sep 29, 2016

(-1/2-sqrt(3)/2i)^3 = 1

Explanation:

de Moivre's theorem tells us that:

(cos theta + i sin theta)^n = cos n theta + i sin n theta

So we find:

(-1/2-sqrt(3)/2i)^3 = (cos (-(2pi)/3) + i sin (-(2pi)/3))^3

color(white)((-1/2-sqrt(3)/2i)^3) = cos (-2pi) + i sin (-2pi)

color(white)((-1/2-sqrt(3)/2i)^3) = 1 + i * 0

color(white)((-1/2-sqrt(3)/2i)^3) = 1

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