Question

Find the value of (1+i)^100.

(a) 2^100 (cos⁡100π+isin100π)

(b) 2^100 (cos⁡25π+isin25π)

(c) 2^50 (cos⁡100π+isin100π)

(d) 2^50 (cos⁡25π+isin25π)

This question was addressed to me during an online exam.

My question is from DeMoivre’s Theorem in chapter Complex Numbers of Engineering Mathematics

Answer 1

Right answer is (d) 2^50 (cos⁡25π+isin25π)

Easiest explanation: We know that,

1+i=\(\sqrt 2 (\frac{1}{\sqrt 2}+\frac{i}{\sqrt 2})=\sqrt 2 (cos \frac{\pi}{4}+isin \frac{\pi}{4})\)

\((1+i)^{100}=(\sqrt 2 \left (cos \frac{\pi}{4}+isin \frac{\pi}{4}\right ))^{100}=2^{50}(\left(cos \frac{\pi}{4}+isin \frac{\pi}{4}\right))^{100}\)

By Applying the DeMoivre’s Theorem

\((1+i)^{100}=2^{50} \left (cos 100\frac{\pi}{4}+isin 100\frac{\pi}{4} \right )\)

\((1+i)^{100}=2^{50} (cos⁡25\pi+isin25\pi)\).