There we go with the first question on complex number 💃🏽💃🏽

$x_1 = 1$, $y_1 = \sqrt{3}$.

Since $x_1 > 0$ and $y_1 > 0$, $\theta_1$ is in the first quadrant.

This value is in the range $(-\pi, \pi]$.

For $z_2 = -1 - i$:

$x_2 = -1$, $y_2 = -1$.

Since $x_2 < 0$ and $y_2 < 0$, $\theta_2$ is in the third quadrant.

The reference angle is $\frac{\pi}{4}$. For the third quadrant, $\theta_2 = -\left(\pi - \frac{\pi}{4}\right) = -\frac{3\pi}{4}$.
This value is in the range $(-\pi, \pi]$.

If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then

Substitute the values from part a):

So,

Using De Moivre's Theorem, if $Z = r(\cos\theta + i\sin\theta)$, then $|Z^n| = r^n$.
Here, $Z = \frac{z_1}{z_2}$, so $r = \sqrt{2}$ and $n=3$.

Using De Moivre's Theorem, $\arg(Z^n) = n\theta$.
Here, $\theta = \frac{13\pi}{12}$ and $n=3$.

The argument must be in the range $(-\pi, \pi]$. We adjust $\frac{13\pi}{4}$ by subtracting multiples of $2\pi$:

Since $\frac{5\pi}{4} > \pi$, we subtract $2\pi$ again:

This value is in the range $(-\pi, \pi