Complex Numbers
Cube Roots of Unity
Premium Question
Grade 11
Question:
If $z^2 + z + 1 = 0$ (so $z \in \{\omega, \omega^2\}$, cube roots of unity), find $\sum_{k=1}^6 \left| z^k + z^{-k} \right|$.
Step-by-Step Solution
Key Concept: Use $z^n + z^{-n} = 2 \cos(2n\pi/3)$ and the periodicity $\omega^3 = 1$. Evaluate the sum of absolute values for $k = 1, 2, 3, 4, 5, 6$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 8