Question

Given $z=(1+i\sqrt{3}{)}^{100}$, then $\frac{RE\left(z\right)}{IM\left(z\right)}$ equals

• A. $2^{100}$
• B. $2^{50}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\sqrt{3}$

Answer 1

Answer: (C)

$\frac{1}{\sqrt{3}}$

Given, $z=(1+i\sqrt{3}{)}^{100}$

$=2^{100}{(cos\frac{\pi }{3}+isin\frac{\pi }{3})}^{100}$
$=2^{100}(cos\frac{100\pi }{3}+isin\frac{100\pi }{3})$
$=2^{100}(-cos\frac{\pi }{3}-isin\frac{\pi }{3})$
$=2^{100}(-\frac{1}{2}-\frac{i\sqrt{3}}{2})$
$\Rightarrow \frac{Re\left(z\right)}{Im\left(z\right)}=\frac{-1/2}{-\sqrt{3}/2}=\frac{1}{\sqrt{3}}$