Question

If \( 1, \omega, \omega^2 \) are the cube roots of unity, then evaluate: \[ (2 - \omega)^2 (2 - \omega^2)^2 (2 - \omega^{10})^2 (2 - \omega^{11})^2 \]

• A. \( -7^4 \)
• B. \( 7^4 \)
• C. \( 7^8 \)
• D. \( -7^8 \)

Hint:

Cube Roots of Unity}
Use the identity: \( 1 + \omega + \omega^2 = 0 \)
Also: \( \omega^3 = 1 \), \( \omega \cdot \omega^2 = 1 \)
Simplify high powers using modulo arithmetic

Answer 1

The Correct Option is B

Note: \( \omega^3 = 1 \Rightarrow \omega^{10} = \omega \), \( \omega^{11} = \omega^2 \) So the expression becomes: \[ [(2 - \omega)(2 - \omega^2)]^2 \cdot [(2 - \omega)(2 - \omega^2)]^2 = [(2 - \omega)(2 - \omega^2)]^4 \] Now, \[ (2 - \omega)(2 - \omega^2) = 4 - 2(\omega + \omega^2) + \omega \omega^2 = 4 - 2(-1) + 1 = 7 \] Thus the expression becomes: \[ 7^4 \]