To find all the cube roots of \( -1 + \sqrt{5}i \), let's proceed by expressing this complex number in polar form and then using De Moivre's theorem.

1. **Convert \( -1 + \sqrt{5}i \) to polar form:**

   For a complex number \( z = x + yi \), we can express it in polar form as:
   \[
   z = r \left( \cos \theta + i \sin \theta \right)
   \]
   where:
   \[
   r = \sqrt{x^2 + y^2}
   \]
   and
   \[
   \theta = \arctan \left( \frac{y}{x} \right).
   \]

   For \( z = -1 + \sqrt{5}i \):
   - \( x = -1 \)
   - \( y = \sqrt{5} \)

   So,
   \[
   r = \sqrt{(-1)^2 + (\sqrt{5})^2} = \sqrt{1 + 5} = \sqrt{6}.
   \]

   Now, calculate \( \theta \):
   \[
   \theta = \arctan \left( \frac{\sqrt{5}}{-1} \right).
   \]
   Since \( -1 + \sqrt{5}i \) lies in the second quadrant, we adjust the angle accordingly:
   \[
   \theta = \pi + \arctan \left( -\frac{\sqrt{5}}{1} \right).
   \]
   Let’s denote this as \( \theta_0 \) for simplicity.

2. **Apply the formula for cube roots:**

   The cube roots of \( z = r \left( \cos \theta + i \sin \theta \right) \) are given by:
   \[
   \sqrt[3]{r} \left( \cos \frac{\theta + 2k\pi}{3} + i \sin \frac{\theta + 2k\pi}{3} \right)
   \]
   for \( k = 0, 1, 2 \).

   - Here, \( \sqrt[3]{r} = \sqrt[3]{\sqrt{6}} = \sqrt[6]{6} \).
   - For \( k = 0, 1, 2 \), we get different values of the angle.

3. **Calculate the cube roots:**

   For each \( k \):
   \[
   z_k = \sqrt[6]{6} \left( \cos \frac{\theta_0 + 2k\pi}{3} + i \sin \frac{\theta_0 + 2k\pi}{3} \right).
   \]

   Now we can substitute the values for each \( k \) to obtain the three distinct cube roots of \( -1 + \sqrt{5}i \).

Would you like a more detailed calculation for each cube root value? 

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Here are some questions to explore further:

1. How does converting a complex number to polar form help in finding its roots?
2. What are the steps to calculate the magnitude and angle for a complex number?
3. How does De Moivre's theorem simplify the process of finding roots of complex numbers?
4. Can the cube roots of a complex number have both real and imaginary components?
5. How does the choice of \( k \) values affect the distinctiveness of roots in complex numbers?

**Tip:** When finding roots of complex numbers, always ensure the angle is correctly placed in the appropriate quadrant for accurate polar representation.

Learn how to find the cube roots of the complex number -1 + √5i using De Moivre's Theorem. This solution involves converting the complex number to polar form, applying the cube root formula, and calculating the distinct roots. Suitable for advanced high school students in Grades 11-12.

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