A) 1
B) 256
C) 0
D) \[{{9}^{3}}\]
Correct Answer: C
Solution :
\[1+i\sqrt{3}=2\left( \frac{1}{2}+i\frac{\sqrt{3}}{2} \right)=2\left[ \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right]=2{{e}^{i\pi /3}}\] \ \[{{(1+i\sqrt{3})}^{9}}={{(2{{e}^{i\pi /3}})}^{9}}={{2}^{9}}.{{e}^{i(3\pi )}}\] \[={{2}^{9}}(\cos 3\pi +i\sin 3\pi )=-{{2}^{9}}\] \ \[a+ib={{(1+i\sqrt{3})}^{9}}=-{{2}^{9}}\]; \ \[b=0\].
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