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: Use the properties of cube roots of unity.
For cube roots of unity, \[ 1+\omega+\omega^2=0 \] Also, \[ \omega^3=1 \] From \[ 1+\omega+\omega^2=0, \] we get \[ 1+\omega^2=-\omega \] and \[ 1+\omega=-\omega^2 \]

Step 2: Simplify \(1-\omega+\omega^2\).
\[ 1-\omega+\omega^2=(1+\omega^2)-\omega \] Since \[ 1+\omega^2=-\omega, \] we get \[ 1-\omega+\omega^2=-\omega-\omega=-2\omega \]

Step 3: Simplify \(1+\omega-\omega^2\).
\[ 1+\omega-\omega^2=(1+\omega)-\omega^2 \] Since \[ 1+\omega=-\omega^2, \] we get \[ 1+\omega-\omega^2=-\omega^2-\omega^2=-2\omega^2 \]

Step 4: Substitute in the given expression.
The given expression is \[ (1-\omega+\omega^2)^5+(1+\omega-\omega^2)^5 \] Substituting, \[ (-2\omega)^5+(-2\omega^2)^5 \] \[ =-32\omega^5-32\omega^{10} \] Now, \[ \omega^5=\omega^3\omega^2=\omega^2 \] and \[ \omega^{10}=\omega^9\omega=(\omega^3)^3\omega=\omega \] Therefore, \[ -32\omega^5-32\omega^{10} = -32\omega^2-32\omega \] \[ =-32(\omega+\omega^2) \] Since \[ \omega+\omega^2=-1, \] we get \[ -32(\omega+\omega^2)=-32(-1)=32 \]

Step 5: Final conclusion.
Therefore, \[ \boxed{32} \]