exponentiation of complex numbers

Question

$u=\cos(\frac{\pi}{5}) + i\sin(\frac{\pi}{5})$, z = $-3+4i$.
I have two questions:
1. Is it true that $|u|=1$ ?
2. Is it ok to say that $|z^4\cdot u^{19}|=||5|^4e^{\phi i}|1|^{19}|=5^4 ?$

Answer 1

1. Yes, since $\sin^2(x)+\cos^2(x)=1$ for all $x$.

2. Yes, since $|zw|=|z|\cdot |w|$ for all complex $z,w$. In particular, $|z|^n=|z^n|$ for all integers $n$.

Answer 2

1) Yes. You can verify it via $|u| = \sqrt{\cos^2(\pi/5) + \sin^2(\pi/5)} = 1$.

2)As $|z| = \sqrt{3^2 + 4^2} = 5$. Yes. You don't state what $\phi$ is. (It should be $\arctan -4/3$) but it doesn't matter as $e^{i\phi} = \cos (\phi) + i\sin(\phi)$ which has norm 1 and "norms out".

Answer 3

1) Yes, $|u|=\sqrt{\cos^2(\frac{\pi}{5}) + \sin^2(\frac{\pi}{5})}=1 \\$

2) Yes, because: $|Z_{1}^{a}\cdot Z_{2}^{b}|=|Z_{1}^{a}|\cdot|Z_{2}^{b}|=|Z_{1}|^a\cdot|Z_{2}|^b$