Question

Find the algebraic form of the following complex number \( (1+i\sqrt{3}) ^{2000} \)

Answer 1

\( \)Find the algebrac form of the following complex number \( (1+i \sqrt{3})^{2000} \)

\( =2^{2000}\left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{2000} \)

\( =2^{2000}\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)^{2000} \)

\( =2^{2000}\left(\cos \frac{2000 \pi}{3}+i \sin \frac{2000 \pi}{3}\right) \)\( =2^{2000}\left(\cos \left(666 \pi+\frac{2 \pi}{3}\right)+i \sin \left(666 \pi+\frac{2 \pi}{3}\right)\right. \)

\( =2^{2000}\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) \)

\( =2^{2000}\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right) \)

\( =-2^{1999}+i 2^{1999} \sqrt{3} \)

Thus \( (1+i \sqrt{3})^{2000}=-2^{1999}+i 2^{1999} \sqrt{3} \)

Find the algebraic form of the following complex number \( (1+i\sqrt{3}) ^{2000} \)