To find the cube roots of $z = 4 \left( \cos \left( \frac{2\pi}{3} \right) + i \sin \left( \frac{2\pi}{3} \right) \right)$, we will use De Moivre's Theorem.

Given $z = r (\cos \theta + i \sin \theta)$, the $n$-th roots are given by: $z^{1/n} = r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right)$ for $k = 0, 1, \ldots, n-1$.

For this problem:

• $r = 4$
• $\theta = \frac{2\pi}{3}$
• $n = 3$

Thus, the cube roots of $z$ are given by: $z^{1/3} = 4^{1/3} \left( \cos \left( \frac{\frac{2\pi}{3} + 2k\pi}{3} \right) + i \sin \left( \frac{\frac{2\pi}{3} + 2k\pi}{3} \right) \right)$

Let's calculate the cube roots for $k = 0, 1, 2$.

When $k = 0$:

$z^{1/3} = 4^{1/3} \left( \cos \left( \frac{\frac{2\pi}{3} + 0}{3} \right) + i \sin \left( \frac{\frac{2\pi}{3} + 0}{3} \right) \right)$ $= 4^{1/3} \left( \cos \left( \frac{2\pi}{9} \right) + i \sin \left( \frac{2\pi}{9} \right) \right)$

When $k = 1$:

$z^{1/3} = 4^{1/3} \left( \cos \left( \frac{\frac{2\pi}{3} + 2\pi}{3} \right) + i \sin \left( \frac{\frac{2\pi}{3} + 2\pi}{3} \right) \right)$ $= 4^{1/3} \left( \cos \left( \frac{8\pi}{9} \right) + i \sin \left( \frac{8\pi}{9} \right) \right)$

When $k = 2$:

$z^{1/3} = 4^{1/3} \left( \cos \left( \frac{\frac{2\pi}{3} + 4\pi}{3} \right) + i \sin \left( \frac{\frac{2\pi}{3} + 4\pi}{3} \right) \right)$ $= 4^{1/3} \left( \cos \left( \frac{14\pi}{9} \right) + i \sin \left( \frac{14\pi}{9} \right) \right)$

Now, let's simplify the expressions.

Simplified Results:

1. For $k = 0$: $z^{1/3} = \sqrt[3]{4} \left( \cos \left( \frac{2\pi}{9} \right) + i \sin \left( \frac{2\pi}{9} \right) \right)$

2. For $k = 1$: $z^{1/3} = \sqrt[3]{4} \left( \cos \left( \frac{8\pi}{9} \right) + i \sin \left( \frac{8\pi}{9} \right) \right)$

3. For $k = 2$: $z^{1/3} = \sqrt[3]{4} \left( \cos \left( \frac{14\pi}{9} \right) + i \sin \left( \frac{14\pi}{9} \right) \right)$

If you have any questions or want more details, please let me know!

Here are 8 related questions for further practice:

1. What is the general form of the nth root of a complex number?
2. How do you convert a complex number from rectangular form to polar form?
3. What is De Moivre's Theorem and how is it used?
4. How do you find the cube roots of a real number using complex numbers?
5. What are the geometric interpretations of complex roots on the complex plane?
6. How do you verify the correctness of the roots of a complex number?
7. What is the principal root of a complex number?
8. How do you find the fourth roots of a complex number?

Tip: When dealing with complex roots, always express the number in polar form for easier computation and apply De Moivre's Theorem for roots and powers.