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

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

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Answer 1 · 2014-10-29T17:53:22

By rewriting in trigonometric form,

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

by De Moivre's Theorem,

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

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

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

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

I hope that this was helpful.

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