Question

If \(1,\omega,\omega^2\) denote the cube roots of unity, then the value of \[ (1-\omega+\omega^2)^5+(1+\omega-\omega^2)^5 \] is

• A. \(32\omega^2\)
• B. \(32\omega\)
• C. \(-32\)
• D. \(32\)

Hint:

For cube roots of unity, always remember the key identities: \[ 1+\omega+\omega^2=0 \] and \[ \omega^3=1. \] These identities simplify most expressions involving \(\omega\).

Answer 1

The Correct Option is D

Step 1: Simplify each bracket.
Using \(1+\omega+\omega^2=0\): \(1-\omega+\omega^2 = -2\omega\) and \(1+\omega-\omega^2 = -2\omega^2\).

Step 2: Compute the sum.
\[(-2\omega)^5 + (-2\omega^2)^5 = -32\omega^5 - 32\omega^{10} = -32\omega^2 - 32\omega = -32(\omega^2+\omega) = -32(-1) = 32.\]
\[\boxed{32}\]