To find the fourth roots of $z = -\sqrt{3} + i$, we need to use the nth root theorem. Here are the steps:

Step 1: Convert $z$ to Polar Form

1. Determine the modulus $r$: $r = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2$

2. Determine the argument $\theta$: Since $z = -\sqrt{3} + i$ is located in the second quadrant (negative real, positive imaginary), we find $\theta$ by calculating: $\tan^{-1}\left(\frac{1}{-\sqrt{3}}\right) = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6}$ However, to place it correctly in the second quadrant, we add $\pi$: $\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$

Thus, the polar form of $z$ is: $z = 2 \left( \cos\frac{5\pi}{6} + i \sin\frac{5\pi}{6} \right)$

Step 2: Apply the nth Root Theorem for Fourth Roots

For the fourth roots, we calculate: $w_k = \sqrt[4]{2} \left( \cos\frac{\frac{5\pi}{6} + 2k\pi}{4} + i \sin\frac{\frac{5\pi}{6} + 2k\pi}{4} \right)$ where $k = 0, 1, 2, 3$.

Would you like a detailed calculation for each root?

Here are five related questions to expand your understanding:

1. How would you find the square root instead of the fourth root for this complex number?
2. What would the polar form of $z = -\sqrt{3} + i$ be if it were located in a different quadrant?
3. Can you explain why we add $\pi$ to adjust the argument in the second quadrant?
4. How would you apply the nth root theorem if the modulus of $z$ were a different value?
5. How can you verify each root obtained from the nth root theorem?

Tip: When converting a complex number to polar form, always check the quadrant to ensure the argument is placed correctly.