Question

Evaluate \(\frac{(1+\sqrt 3 i) ^{16}}{(\sqrt 3-i) ^{17}}\).

(a) \(\frac{1}{2}×(cos \frac{\pi}{6}+isin \frac{\pi}{6})\)

(b) \(\frac{1}{3}×(cos \frac{\pi}{6}+isin \frac{\pi}{6})\)

(c) \(\frac{1}{2}×(cos \frac{\pi}{3}+isin \frac{\pi}{3})\)

(d) \(\frac{1}{4}×(cos \frac{\pi}{6}+isin \frac{\pi}{6})\)

The question was posed to me during an interview.

This interesting question is from DeMoivre’s Theorem topic in section Complex Numbers of Engineering Mathematics

Answer 1

Correct option is (a) \(\frac{1}{2}×(cos \frac{\pi}{6}+isin \frac{\pi}{6})\)

Explanation: We know that,

\(1+\sqrt 3 i=2(cos \frac{\pi}{3}+isin \frac{\pi}{3})\)

\(\sqrt 3-i=2(cos \frac{\pi}{6}-isin \frac{\pi}{6})\)

\(\frac{(1+\sqrt 3 i) ^{16}}{(\sqrt 3-i) {^17}}=\frac{2^{16}(cos \frac{\pi}{3}+isin \frac{\pi}{3}) ^{16}}{2^{17}(cos \frac{\pi}{6}-isin \frac{\pi}{6}) ^{17}}\)

By Demoivre’s Theorem

= \(\frac{1}{2}×(cos \frac{16\pi}{3}+isin \frac{16\pi}{3})(cos \frac{17\pi}{6}+isin \frac{17\pi}{6})\)

= \(\frac{1}{2}×(cos(\frac{⁡49\pi}{6})+isin(\frac{49\pi}{6}))\)

= \(\frac{1}{2}×(cos\frac{\pi}{6}+isin\frac{\pi}{6})\).