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.