To evaluate \((12 + 3\sqrt{2}i)^{100}\), we will use De Moivre's Theorem, which involves converting the complex number to polar form.

### Step 1: Express in polar form
A complex number in rectangular form, \( z = a + bi \), can be written in polar form as:
\[
z = r(\cos \theta + i \sin \theta)
\]
where:
- \( r = \sqrt{a^2 + b^2} \) is the modulus,
- \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \) is the argument.

Here, \( a = 12 \) and \( b = 3\sqrt{2} \).

1. **Find \( r \):**
\[
r = \sqrt{12^2 + (3\sqrt{2})^2} = \sqrt{144 + 18} = \sqrt{162} = 9\sqrt{2}
\]

2. **Find \( \theta \):**
\[
\theta = \tan^{-1}\left(\frac{3\sqrt{2}}{12}\right) = \tan^{-1}\left(\frac{\sqrt{2}}{4}\right)
\]

### Step 2: Apply De Moivre's Theorem
De Moivre's Theorem states:
\[
z^n = r^n \left( \cos(n\theta) + i \sin(n\theta) \right)
\]
For \( z = 12 + 3\sqrt{2}i \) and \( n = 100 \), we apply this theorem using the calculated values for \( r \) and \( \theta \).

1. **Compute \( r^{100} \):**
\[
r^{100} = (9\sqrt{2})^{100}
\]

2. **Compute \( \theta_{100} = 100\theta \).**

Finally, the result will be in the form of \( r^{100} (\cos(100\theta) + i \sin(100\theta)) \).

Would you like a more detailed breakdown of the steps, or assistance with further simplifications? Let me know!

Here are five related questions to explore:
1. How does De Moivre's Theorem apply to other exponents of complex numbers?
2. What is the geometric meaning of multiplying complex numbers in polar form?
3. How can polar form simplify complex number exponentiation?
4. What are the properties of complex roots of unity?
5. How does converting between rectangular and polar forms affect the visualization of complex numbers?

**Tip:** Always simplify complex expressions by converting them to polar form for easier exponentiation!

Learn how to solve the complex number exponentiation (12 + 3√2i)^100 by applying De Moivre's Theorem. This problem explores complex numbers in polar form, their modulus, argument, and the application of exponentiation. Suitable for undergraduate students studying complex analysis.

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