Question

Simplified form of $\frac{(\sqrt{3}-i{)}^6}{(1+i{)}^8}$ is

• A. $4i$
• B. $4-4i$
• C. $-4$
• D. $0$

Answer 1

The correct option is C $-4$

$\because \sqrt{3}-i=2(\cos \frac{\pi }{6}-i\sin \frac{\pi }{6})$
$\therefore {(\sqrt{3}-i)}^6=64(\cos \pi -i\sin \pi )=-64$
Similarly, $1+i=\sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$
$\therefore (1+i{)}^8=16(\cos 2\pi +i\sin 2\pi )=16$
So, $\frac{(\sqrt{3}-i{)}^6}{(1+i{)}^8}=-\frac{64}{16}$
$=-4$