Question

If $\omega$ is an imaginary cube root of unity, then $(1+\omega-\omega^2)^7$ is equal to

• A. $128\omega$
• B. $-128\omega$
• C. $128\omega^2$
• D. $-128\omega^3$
• E. $-128\omega^2$

Hint:

Always reduce powers of $\omega$ using $\omega^3=1$.

Answer 1

The Correct Option is C

Step 1: Use identity. \[ 1+\omega+\omega^2 = 0 \Rightarrow 1+\omega = -\omega^2 \]

Step 2: Simplify expression. \[ 1+\omega-\omega^2 = (1+\omega) - \omega^2 \] \[ = -\omega^2 - \omega^2 = -2\omega^2 \]

Step 3: Raise to power 7. \[ (-2\omega^2)^7 = (-2)^7 (\omega^2)^7 \] \[ = -128 \cdot \omega^{14} \]

Step 4: Reduce power of $\omega$. \[ \omega^3 = 1 \Rightarrow \omega^{14} = \omega^2 \] \[ = -128 \omega^2 \] \[ \boxed{-128\omega^2} \]