If (1+i 3)^9=a+i b, then b is equal to - Vidyadip

MCQ

If $(1+i \sqrt{3})^9=a+i b$, then $b$ is equal to

A
1
B
256
✓
$0$
D
$2^9$

✓

Answer

Correct option: C.

$0$

(C)
$1+ i \sqrt{3}=2\left(\frac{1}{2}+ i \frac{\sqrt{3}}{2}\right)=2\left[\cos \frac{\pi}{3}+ i \sin \frac{\pi}{3}\right]=2 e ^{\frac{ i \pi}{3}}$
$\therefore \quad(1+ i \sqrt{3})^9=\left(2 e ^{ i \pi / 3}\right)^9=2^9 \cdot e ^{ i (3 \pi)}$
$=2^9(\cos 3 \pi+i \sin 3 \pi)$
$=-2^9$
$\therefore \quad a + ib =(1+ i \sqrt{3})^9=-2^9$
$\therefore b =0$

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