Complex Numbers
Cube Roots of Unity
Premium Question
Grade 11
Question:
If $\omega \neq 1$ is a cube root of unity, then $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$ equals:
(1) $0$
(2) $16$
(3) $32$
(4) $-32$
Step-by-Step Solution
Key Concept: Use $1 + \omega + \omega^2 = 0$ to write $1 - \omega + \omega^2 = -2\omega$ and $1 + \omega - \omega^2 = -2\omega^2$. Then apply $\omega^3 = 1$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)