
\begin{enumerate}
\item $|z_1| = |1+\i|^4 = \big(\sqrt2\big)^4 = 4$.

$\arg(z_1) = \arg\big[(1+\i)^4\big] = 4\arg(1+\i) = 4\times\dfrac{\pi}{4}\mod2\pi = \pi\mod2\pi$.

Donc $z_1 = 4\big(\cos(\pi)+\i\sin(\pi)\big)=4(-1+\i\times0)$, soit $\boxed{z_1=-4}$.

\item $\left|\dfrac{1+\i}{1-\i}\right|=\dfrac{|1+\i|}{|1-\i|}=1$.

$\arg\left(\dfrac{1+\i}{1-\i}\right) = \arg(1+\i) - \arg(1-\i) = \dfrac{\pi}{4}-\left(-\dfrac{\pi}{4}\right)=\dfrac{\pi}{2}\mod2\pi$.

On en déduit alors que $z_2=\cos\dfrac{\pi}{2}+\i\sin\dfrac{\pi}{2}$, soit $\boxed{z_2=\i}$.
\end{enumerate}
