32(cos60^circ +isin60^circ ) Next, find Theta 's for all 5 roots of 16+16sqrt (3)i Theta =60^circ ,420^circ ,[?]^circ ,square ^circ ,square ^circ Remember, to find the different Theta 's add 360^circ to Theta

To find the roots of the complex number $16 + 16\sqrt{3}i$, we first need to express it in polar form.

1. **Convert to Polar Form:**

The magnitude $r$ of the complex number $16 + 16\sqrt{3}i$ is:
$$r = \sqrt{16^2 + (16\sqrt{3})^2} = \sqrt{256 + 768} = \sqrt{1024} = 32$$

The argument $\theta$ is:
$$\theta = \tan^{-1}\left(\frac{16\sqrt{3}}{16}\right) = \tan^{-1}(\sqrt{3}) = 60^\circ$$

So, the complex number in polar form is:
$$32(\cos 60^\circ + i \sin 60^\circ)$$

2. **Find the Roots:**

The roots of the complex number $32(\cos 60^\circ + i \sin 60^\circ)$ are given by:
$$z_k = r^{1/n} \left( \cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n} \right)$$
where $n$ is the number of roots, and $k = 0, 1, 2, \ldots, n-1$.

For $16 + 16\sqrt{3}i$, $n = 5$ (since it is a 5th degree polynomial).

The roots are:
$$z_k = 32^{1/5} \left( \cos \frac{60^\circ + 360^\circ k}{5} + i \sin \frac{60^\circ + 360^\circ k}{5} \right)$$

Simplifying $32^{1/5}$:
$$32^{1/5} = 2$$

So the roots are:
$$z_k = 2 \left( \cos \frac{60^\circ + 360^\circ k}{5} + i \sin \frac{60^\circ + 360^\circ k}{5} \right)$$

3. **Calculate the Specific Roots:**

For $k = 0$:
$$z_0 = 2 \left( \cos \frac{60^\circ}{5} + i \sin \frac{60^\circ}{5} \right) = 2 \left( \cos 12^\circ + i \sin 12^\circ \right)$$

For $k = 1$:
$$z_1 = 2 \left( \cos \frac{420^\circ}{5} + i \sin \frac{420^\circ}{5} \right) = 2 \left( \cos 84^\circ + i \sin 84^\circ \right)$$

For $k = 2$:
$$z_2 = 2 \left( \cos \frac{780^\circ}{5} + i \sin \frac{780^\circ}{5} \right) = 2 \left( \cos 156^\circ + i \sin 156^\circ \right)$$

For $k = 3$:
$$z_3 = 2 \left( \cos \frac{1140^\circ}{5} + i \sin \frac{1140^\circ}{5} \right) = 2 \left( \cos 228^\circ + i \sin 228^\circ \right)$$

For $k = 4$:
$$z_4 = 2 \left( \cos \frac{1320^\circ}{5} + i \sin \frac{1320^\circ}{5} \right) = 2 \left( \cos 264^\circ + i \sin 264^\circ \right)$$

So, the five roots are:
$$\Theta = 60^\circ, 420^\circ, 12^\circ, 84^\circ, 156^\circ$$

Adding $360^\circ$ to each $\Theta$ as needed:
$$\Theta = 60^\circ, 420^\circ, 372^\circ, 444^\circ, 516^\circ$$

Thus, the complete list of $\Theta$ values for the roots is:
\[
60^\circ, 420^\circ, 372^\circ, 444^\circ, 516^\circ