Complex Numbers
Cube Roots of Unity – Powers
Complex Numbers_PYQ
Grade 11
Question:
If $\omega$ is an imaginary cube root of unity, then $(1 + \omega - \omega^2)^7$ is equal to
$128\omega$
$-128\omega$
$128\omega^2$
$-128\omega^2$
Step-by-Step Solution
Key Concept: Using $1+\omega=-\omega^2$ to collapse the bracket to $-2\omega^2$, then $\omega^{14}=\omega^{14\bmod3}=\omega^2$.
**Step 1: Simplify the bracket**
$1+\omega+\omega^2=0 \Rightarrow 1+\omega=-\omega^2$. So $1+\omega-\omega^2 = -\omega^2-\omega^2 = -2\omega^2$.
**Step 2: Raise to 7th power**
$(-2\omega^2)^7 = -2^7\cdot\omega^{14} = -128\cdot\omega^{14\bmod 3} = -128\omega^2$.
Correct Answer: 4