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Let's work through the exercises step by step. I'll begin with Exercise 1, part (a), and solve the indicated roots using De Moivre's Theorem.

**1. Hallar las raíces indicadas usando el teorema de Moivre**

**a. Las raíces cúbicas de 1:**
To find the cubic roots of 1, we first express 1 in polar form. The polar form of 1 is:
\[
1 = \text{cis} 0^\circ = 1(\cos(0^\circ) + i \sin(0^\circ))
\]

The general formula for finding the \(n\)-th roots of a complex number is:
\[
z_k = r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right)
\]
where \(k = 0, 1, 2, \dots, n-1\).

For \(z^3 = 1\), we have:
- \(r = 1\)
- \(\theta = 0\)

Now, using De Moivre's formula for the cubic roots:
\[
z_k = \cos \left( \frac{2k\pi}{3} \right) + i \sin \left( \frac{2k\pi}{3} \right)
\]
For \(k = 0, 1, 2\):
1. \(k = 0\): \(\cos(0^\circ) + i \sin(0^\circ) = 1\)
2. \(k = 1\): \(\cos \left( \frac{2\pi}{3} \right) + i \sin \left( \frac{2\pi}{3} \right) = -\frac{1}{2} + i \frac{\sqrt{3}}{2}\)
3. \(k = 2\): \(\cos \left( \frac{4\pi}{3} \right) + i \sin \left( \frac{4\pi}{3} \right) = -\frac{1}{2} - i \frac{\sqrt{3}}{2}\)

So, the cubic roots of 1 are:
\[
1, \, -\frac{1}{2} + i \frac{\sqrt{3}}{2}, \, -\frac{1}{2} - i \frac{\sqrt{3}}{2}
\]

Would you like to proceed with part (b) or ask for further details on this? Let me know!

Here are some questions for further exploration:
1. How does the general formula for the \(n\)-th roots of a complex number work?
2. What is the geometric interpretation of the cubic roots on the complex plane?
3. Can you derive the square roots of a different number, such as \(i\)?
4. How does De Moivre's Theorem extend to higher powers of complex numbers?
5. What are the applications of complex roots in physics and engineering?

**Tip:** When solving for roots of complex numbers, it’s helpful to visualize them on the unit circle to better understand their angles and symmetries.

Hallar las raíces indicadas usando el teorema de Moivre y convertir entre notación trigonométrica y rectangular.

Solve problems involving complex roots using De Moivre's Theorem, convert between rectangular and trigonometric forms, and practice working with complex numbers. Detailed step-by-step solutions included.

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