The equation in the image is:

$z^4 = 8 + 8\sqrt{3}i$

This is a complex number equation that we need to solve for $z$. We can solve it using polar form and De Moivre's Theorem. Here's the step-by-step process:

Step 1: Express the right-hand side in polar form

The complex number $8 + 8\sqrt{3}i$ can be written in polar form $r(\cos \theta + i \sin \theta)$, where:

• $r$ is the modulus,
• $\theta$ is the argument (angle).

Modulus:

$r = \sqrt{8^2 + (8\sqrt{3})^2} = \sqrt{64 + 192} = \sqrt{256} = 16$

Argument:

$\theta = \tan^{-1} \left( \frac{8\sqrt{3}}{8} \right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$

So the polar form is: $16 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$

Step 2: Apply De Moivre's Theorem

To solve $z^4 = 16 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$, we take the fourth root of both sides.

The fourth root of the modulus $r = 16$ is: $\sqrt[4]{16} = 2$

The argument is divided by 4: $\frac{\frac{\pi}{3} + 2k\pi}{4} = \frac{\pi}{12} + \frac{k\pi}{2}, \quad k = 0, 1, 2, 3$

So the four solutions for $z$ are:

$z_k = 2 \left( \cos \left( \frac{\pi}{12} + \frac{k\pi}{2} \right) + i \sin \left( \frac{\pi}{12} + \frac{k\pi}{2} \right) \right), \quad k = 0, 1, 2, 3$

These are the four roots in polar form.

Would you like to see the detailed steps for each solution?

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Here are 5 related questions to explore this topic further:

1. What is De Moivre's Theorem and how is it used in solving complex number equations?
2. How do you convert a complex number from rectangular form to polar form?
3. What are the different methods for solving higher-degree equations in complex numbers?
4. How does the modulus and argument of a complex number affect its polar representation?
5. How can you geometrically interpret the four roots of the equation $z^4 = 8 + 8\sqrt{3}i$?

Tip: Always check the quadrant of the complex number to get the correct argument $\theta$ when converting to polar form.