Question

simplify ((1/2) +(sqrt(3/2)*i))^6

Answer 1

$\underline{\textsf{De Moivre's theorem :}}$

$\textsf{For any positive or negative integer n,}$
$\textsf{and real value of}\:\theta$

$\small{(cos\theta+ isin\theta)^{n}=cos\:n\theta+isin\:n\theta}$

$\boxed{\underline{\textsf{Solution :}}}$

$\textsf{Now,}\:\bigg(\frac{1}{2}+i\frac{\sqrt{3}}{2}\bigg)^{6}$

$=\bigg(cos\frac{\pi}{3}+isin\frac{\pi}{3}\bigg)^{6}$

$=cos\frac{6\pi}{3}+isin\frac{6\pi}{3}$

$\textsf{by De Moivre's theorem}$

$=cos2\pi + isin2\pi$

$=cos0+isin0$

$=1+0$

$=\bold{1}$

$\to \boxed{\boxed{\bold{\bigg(\frac{1}{2}+i\frac{\sqrt{3}}{2}\bigg)^{6}=1}}}$