Question

$a+ib=(1+i\sqrt{3}{)}^{300}$ then $a=$ _____ and $b=$ ______

• A. $0,1$
• B. $2^{300},0$
• C. $0,0$
• D. none of these

Answer 1

Answer: (B)

$2^{300},0$

$a+ib=(1+i\sqrt{3}{)}^{300}$

Let $z=1+i\sqrt{3}$

$z=2[\frac{1}{2}+i\frac{\sqrt{3}}{2}]$

$z^{300}=2{[\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}]}^{300}$

$z=(2{)}^{300}[\cos 300\frac{\pi }{3}+i\sin 300\frac{\pi }{3}]$

$=\left(2{)}^{300}\right[\cos 100\pi +i\sin 100\pi ]$

$=\left(2{)}^{300}\right[1+0i]$

$=(2{)}^{300}$

$a+ib=(2{)}^{300}$

Thus $a=(2{)}^{300}$ and $b=0$