To find the cube roots of a complex number such as $z = -1 + i$, we can follow these steps:

1. Convert the complex number into polar form:
   A complex number $z = a + bi$ can be expressed as $z = r (\cos \theta + i \sin \theta)$, where $r$ is the modulus and $\theta$ is the argument (angle).

   • Modulus $r$ is given by:
     $r = \sqrt{a^2 + b^2} = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$

   • Argument $\theta$ is the angle formed with the positive real axis:
     $\theta = \tan^{-1} \left( \frac{b}{a} \right) = \tan^{-1} \left( \frac{1}{-1} \right) = \tan^{-1} (-1) = \frac{3\pi}{4}$ (We use $\frac{3\pi}{4}$ because $-1 + i$ is in the second quadrant.)

   Therefore, the polar form of $-1 + i$ is:
   $z = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right)$

2. Find the cube roots:
   The formula for the $n$-th roots of a complex number in polar form is:
   $z^{1/n} = r^{1/n} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right) \quad \text{for} \, k = 0, 1, \dots, n-1$ Here, $n = 3$, $r = \sqrt{2}$, and $\theta = \frac{3\pi}{4}$.

   Thus, the cube roots are:
   $z_k = \sqrt[3]{\sqrt{2}} \left( \cos \frac{3\pi/4 + 2k\pi}{3} + i \sin \frac{3\pi/4 + 2k\pi}{3} \right) \quad \text{for} \, k = 0, 1, 2$

3. Calculate each root:

   • For $k = 0$:
     [ z_0 = \sqrt[6]{2} \left( \cos \frac{3\pi}{12} + i \sin \frac{3\pi}{12} \right) = \sqrt[6]{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) ] $z_0 = \sqrt[6]{2} \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right)$

   • For $k = 1$:
     $z_1 = \sqrt[6]{2} \left( \cos \frac{11\pi}{12} + i \sin \frac{11\pi}{12} \right)$

   • For $k = 2$:
     $z_2 = \sqrt[6]{2} \left( \cos \frac{19\pi}{12} + i \sin \frac{19\pi}{12} \right)$

Thus, the three cube roots of $-1 + i$ are given by the above expressions for $z_0$, $z_1$, and $z_2$.

Would you like a more detailed explanation of any of these steps?

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Related questions:

1. What is the process to find the nth roots of a complex number in general?
2. How can we convert a complex number from rectangular form to polar form?
3. What is the geometric interpretation of complex roots on the complex plane?
4. How do cube roots of unity relate to finding roots of complex numbers?
5. What are some real-world applications of finding roots of complex numbers?

Tip: Always express complex numbers in polar form when dealing with powers and roots—it simplifies the process.