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The cube roots of the complex number \((-1 + i)\) are:

1. \( 0.7937 + 0.7937i \)
2. \( -1.0842 + 0.2905i \)
3. \( 0.2905 - 1.0842i \)

These are the three possible values for \(\left( -1 + i \right)^{1/3}\), as cube roots for complex numbers yield three distinct solutions due to the periodic nature of the complex exponential function.

Would you like further details on how these roots were calculated, or do you have any questions?

Here are 5 related questions to expand this topic:
1. How do we calculate higher roots of complex numbers?
2. What is the geometric interpretation of complex roots in the Argand plane?
3. Why do cube roots of complex numbers have multiple solutions?
4. How are cube roots used in solving cubic equations involving complex numbers?
5. Can these complex cube roots be represented in polar form?

**Tip:** When working with complex numbers, expressing them in polar form (magnitude and angle) can simplify root calculations significantly!

Explore how to find the cube roots of the complex number (-1 + i) with a detailed solution. The problem covers key concepts like De Moivre's Theorem, Euler's formula, and the periodic nature of complex roots. Suitable for advanced high school and college-level students.

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