Question

If \( \omega \) is an imaginary cube root of unity, find the value of \( (1 + \omega - \omega^2)^7 \).

• A. \(128\omega^2\)
• B. \(-128\omega^2\)
• C. \(64\omega\)
• D. \(-64\omega^2\)

Hint:

For cube roots of unity always remember: \[ 1+\omega+\omega^2=0,\qquad \omega^3=1 \] Reduce higher powers using \( \omega^3=1 \) and simplify expressions quickly.

Answer 1

The Correct Option is B

Concept: For cube roots of unity, \[ 1 + \omega + \omega^2 = 0 \] and \[ \omega^3 = 1 \] These identities allow simplification of expressions involving powers of \( \omega \).

Step 1: Use the identity of cube roots of unity. \[ 1 + \omega + \omega^2 = 0 \] Rearranging, \[ 1 + \omega = -\omega^2 \]

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

Step 3: Raise to the power \(7\). \[ (1 + \omega - \omega^2)^7 = (-2\omega^2)^7 \] \[ = (-2)^7 (\omega^2)^7 \] \[ = -128\,\omega^{14} \]

Step 4: Reduce the power of \( \omega \). Since \[ \omega^3 = 1 \] \[ \omega^{14} = \omega^{12+2} = (\omega^3)^4\omega^2 \] \[ = 1^4 \omega^2 = \omega^2 \] Thus, \[ (1 + \omega - \omega^2)^7 = -128\omega^2 \] \[ \boxed{-128\omega^2} \]