What is $(-\frac{1}{2}+\frac{i\sqrt{3}}{2})^{10}$ ?

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Answer 1 · 2014-10-29T17:53:22

By rewriting in trigonometric form,

$(-1/2+sqrt{3}/2i)^{10}=[cos({2pi}/3)+i sin({2pi}/3)]^{10}$

by De Moivre's Theorem,

$=cos(10cdot{2pi}/3)+i sin(10cdot{2pi}/3)$

$=cos({20pi}/3)+i sin({20pi}/3)$

$=cos({2pi}/3)+i sin({2pi}/3)$

$=-1/2+sqrt{3}/2i$

I hope that this was helpful.

What is (-\frac{1}{2}+\frac{i\sqrt{3}}{2})^{10}(-\frac{1}{2}+\frac{i\sqrt{3}}{2})^{10}?