Question

Given $z=(1+i\sqrt{3}{)}^{100}$, then $\left[\frac{Re\left(z\right)}{Im\left(z\right)}\right]$ equals

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

Answer 1

Answer: (C)

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

$z=(1+i\sqrt{3}{)}^{100}$
$z=(2e^{i\frac{\pi }{3}}{)}^{100}$
By DeMoivre's Theorem,
$z=2^{100}{e}^{i\frac{100\pi }{3}}$
$z=2^{100}{e}^{i\frac{4\pi }{3}}$
Real part $=-\frac{1}{2}$
Imaginary part $=-\frac{\sqrt{3}}{2}$
Hence, option C is correct.