The exercise you provided is asking to find the roots of several complex numbers using De Moivre's Theorem. Here's an outline of how to solve these types of problems:

De Moivre's Theorem:

For a complex number in polar form, $z = r(\cos \theta + i \sin \theta)$, the $n$-th roots are given by: $z_k = r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right), \quad k = 0, 1, \dots, n-1$ Where $r$ is the modulus and $\theta$ is the argument of the complex number.

Let's look at some of the problems you shared:

c. Las raíces cuartas de $1 - \sqrt{3} i$

1. Convert to polar form:

   • $r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$
   • $\theta = \tan^{-1} \left( \frac{-\sqrt{3}}{1} \right) = -\frac{\pi}{3}$ So, $1 - \sqrt{3}i = 2 \left( \cos \left( -\frac{\pi}{3} \right) + i \sin \left( -\frac{\pi}{3} \right) \right)$.

2. Find the fourth roots ($n = 4$): The fourth roots are given by: $z_k = 2^{1/4} \left( \cos \left( \frac{-\pi/3 + 2k\pi}{4} \right) + i \sin \left( \frac{-\pi/3 + 2k\pi}{4} \right) \right), \quad k = 0, 1, 2, 3$ You can calculate the values of these roots for $k = 0, 1, 2, 3$.

d. Las raíces cuadradas de $1 + i$

1. Convert to polar form:

   • $r = \sqrt{1^2 + 1^2} = \sqrt{2}$
   • $\theta = \tan^{-1} \left( \frac{1}{1} \right) = \frac{\pi}{4}$ So, $1 + i = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$.

2. Find the square roots ($n = 2$): The square roots are: $z_k = (\sqrt{2})^{1/2} \left( \cos \left( \frac{\pi/4 + 2k\pi}{2} \right) + i \sin \left( \frac{\pi/4 + 2k\pi}{2} \right) \right), \quad k = 0, 1$ Solve for $k = 0, 1$.

Would you like to explore the full solutions step-by-step or go through another part of the exercise?

Here are 5 related questions to think about:

1. How do you find the modulus and argument of a complex number in polar form?
2. What are the main steps of De Moivre's Theorem?
3. How can you use Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$ for roots of complex numbers?
4. What is the geometrical interpretation of complex roots in the complex plane?
5. How does the number of roots relate to the angle between each root?

Tip: When finding roots of a complex number, always express the number in polar form first, as this simplifies the process significantly!