To solve \((\sqrt{3} - i)^4\), we can use De Moivre's Theorem, which is useful when raising complex numbers to a power. Here's the step-by-step process:

1. **Convert the complex number to polar form**:  
   A complex number \( z = a + bi \) can be represented in polar form as:
   \[
   z = r(\cos \theta + i \sin \theta)
   \]
   where \( r = \sqrt{a^2 + b^2} \) is the magnitude and \( \theta = \tan^{-1} \left(\frac{b}{a}\right) \) is the argument.

   For \( \sqrt{3} - i \):
   - \( a = \sqrt{3} \)
   - \( b = -1 \)
   \[
   r = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2
   \]
   \[
   \theta = \tan^{-1}\left(\frac{-1}{\sqrt{3}}\right) = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right)
   \]

   Since \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \), and the point lies in the fourth quadrant:
   \[
   \theta = -\frac{\pi}{6}
   \]
   So, in polar form:
   \[
   \sqrt{3} - i = 2 \left( \cos\left(-\frac{\pi}{6}\right) + i \sin\left(-\frac{\pi}{6}\right) \right)
   \]

2. **Apply De Moivre's Theorem**:  
   \[
   z^n = r^n \left( \cos(n\theta) + i\sin(n\theta) \right)
   \]
   Substituting \( r = 2 \), \( \theta = -\frac{\pi}{6} \), and \( n = 4 \):
   \[
   (\sqrt{3} - i)^4 = 2^4 \left( \cos\left(4 \times -\frac{\pi}{6}\right) + i \sin\left(4 \times -\frac{\pi}{6}\right) \right)
   \]
   \[
   = 16 \left( \cos\left(-\frac{4\pi}{6}\right) + i \sin\left(-\frac{4\pi}{6}\right) \right)
   \]
   Simplify the angle:
   \[
   = 16 \left( \cos\left(-\frac{2\pi}{3}\right) + i \sin\left(-\frac{2\pi}{3}\right) \right)
   \]

3. **Evaluate the trigonometric functions**:  
   \[
   \cos\left(-\frac{2\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(-\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2}
   \]
   Therefore:
   \[
   (\sqrt{3} - i)^4 = 16 \left( -\frac{1}{2} - i\frac{\sqrt{3}}{2} \right)
   \]
   \[
   = 16 \left( -\frac{1}{2} - i\frac{\sqrt{3}}{2} \right) = -8 - 8\sqrt{3}i
   \]

So, the result is:
\[
(\sqrt{3} - i)^4 = -8 - 8\sqrt{3}i
\]

Would you like further details or have any questions?

Here are five questions related to this topic:
1. How does De Moivre's Theorem help in simplifying powers of complex numbers?
2. What is the significance of the polar form of a complex number?
3. How do you determine which quadrant a complex number lies in?
4. What are the steps to convert a complex number from rectangular to polar form?
5. How does the argument of a complex number change when raised to a power?

**Tip:** When working with complex numbers, always check the quadrant to ensure correct angle measurements.

Learn how to solve the complex number problem (sqrt3-i)^4 using De Moivre's Theorem with a step-by-step explanation. This problem explores the application of De Moivre's Theorem in simplifying powers of complex numbers. Suitable for advanced high school students and above.

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