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:

Properties of cube roots of unity: \(1+\omega+\omega^2=0\) and \(\omega^3=1\).
Use these properties to simplify expressions involving \(\omega\) and \(\omega^2\) before raising to a power.
E.g., \(1+\omega^2 = -\omega\) and \(1+\omega = -\omega^2\).

Answer 1

The Correct Option is D

Properties of cube roots of unity: 1. \(1 + \omega + \omega^2 = 0\) 2. \(\omega^3 = 1\) From (1), we can derive: \(1 + \omega^2 = -\omega\) \(1 + \omega = -\omega^2\) Consider the first term: \((1-\omega+\omega^2)^5\). Substitute \(1+\omega^2 = -\omega\): \(( (1+\omega^2) - \omega )^5 = (-\omega - \omega)^5 = (-2\omega)^5 = (-2)^5 \omega^5 = -32 \omega^5\). Since \(\omega^3 = 1\), then \(\omega^5 = \omega^3 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2\). So, the first term is \(-32\omega^2\). Consider the second term: \((1+\omega-\omega^2)^5\). Substitute \(1+\omega = -\omega^2\): \(( (1+\omega) - \omega^2 )^5 = (-\omega^2 - \omega^2)^5 = (-2\omega^2)^5 = (-2)^5 (\omega^2)^5 = -32 \omega^{10}\). Since \(\omega^3 = 1\), then \(\omega^{10} = (\omega^3)^3 \cdot \omega = 1^3 \cdot \omega = \omega\). So, the second term is \(-32\omega\). Now, sum the two terms: \( (1-\omega+\omega^2)^5 + (1+\omega-\omega^2)^5 = -32\omega^2 - 32\omega \) \( = -32(\omega^2 + \omega) \). From \(1 + \omega + \omega^2 = 0\), we have \(\omega + \omega^2 = -1\). So, the sum is \(-32(-1) = 32\). This matches option (d). \[ \boxed{32} \]