A power (!)

Evaluate

$$\mathcal{P}= \left ( \frac{1+\sqrt{3}}{2\sqrt{2}}+ i \frac{\sqrt{3}-1}{2\sqrt{2}} \right )^{72}$$

Solution

\begin{align*} \left ( \frac{1+\sqrt{3}}{2\sqrt{2}}+ i \frac{\sqrt{3}-1}{2\sqrt{2}} \right )^{72} &=  \left [ \left ( \frac{1}{2}\cdot \frac{\sqrt{2}}{2}+ \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2} \right )+ i \left ( \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2} -\frac{1}{2}\frac{\sqrt{2}}{2} \right ) \right ]^{72}\\
&=\biggl [ \left ( \cos \frac{\pi}{3}\cos \frac{\pi}{4} + \sin \frac{\pi}{3}\sin \frac{\pi}{4} \right ) +\\& \qquad \qquad +i \left ( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4}  \right ) \biggr ]^{72} \\
&= \left [ \cos \left ( \frac{\pi}{3} - \frac{\pi}{4} \right ) +i \sin \left ( \frac{\pi}{3} - \frac{\pi}{4} \right )\right ]^{72}\\
&= \left ( \cos \frac{\pi}{12} + i \sin \frac{\pi}{12} \right )^{72}\\ &= \left ( \cos^2 \frac{\pi}{12} - \sin^2 \frac{\pi}{12} +2i \cos \frac{\pi}{12} \sin \frac{\pi}{12} \right )^{36}\\ &=\left ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right )^{36} \\
&=\left ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right )^{36}\\
&=\left ( \frac{\sqrt{3}}{2}+\frac{i}{2} \right )^{36} \\
&=\left [ \left ( \frac{\sqrt{3}}{2}+ \frac{i}{2} \right )^3 \right ]^{12}\\
&=1\end{align*}