FORM GR1268 Question 27

Question:
\((1+i)^{10}=\)

(A) \(1\)     (B) \(i\)     (C) \(32\)     (D) \(32i\)     (E) \(32(i+1)\)

Answer:
(D)

Answer Key:
\[\begin{align}
(1+i)^{10}&=\left(\sqrt2 e^{\frac{\pi}{4}i}\right)^{10}\\
&=2^{\frac{10}{2}}e^{\frac{10\pi}{4}i}\\
&=32e^{\frac{5\pi}{2}i}\\
&=32\left(\cos\frac{5\pi}{2}+i\sin\frac{5\pi}{2}\right)\\
&=32\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)\\
&=32(0+i)\\
&=32i.
\end{align}\]