<s>[INST] Let z = 1 + i. Find the real and imaginary parts of z^19 Let z = 1 + i. Find the real and imaginary parts of $z^{19}$
 [/INST] It is quickly determined that $\frac{z}{\sqrt{2}} = \frac{1+i}{\sqrt{2}} = e^{\pi i/4}$ and 
\begin{align}
\left(\frac{z}{\sqrt{2}} \right)^{19} &= e^{19 \pi i/4} = e^{4 \pi i + 3 \pi i/4} = \cos(4\pi) \, \left( \cos\left(\frac{3 \pi}{4}\right) + i \, \sin\left(\frac{3\pi }{4} \right) \right) \\
&= \frac{-1 + i}{\sqrt{2}}. 
\end{align}
From this: $z^{19} = 2^{9} \, (-1 + i)$.
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