https://file.aiquickdraw.com/mathbot/file/1728286111237f1jid81e.jpg

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.

This solution explains how to solve the complex equation z^4 = 8 + 8√3i using polar form and De Moivre's Theorem. Step-by-step, it covers converting the complex number into polar coordinates, finding the modulus and argument, and applying De Moivre's Theorem to find the four roots.

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