Question

Let \( \omega \ne 1 \) be a cube root of unity and \( (1+\omega)^7 = a + \omega \). Then the value of \( a \) is

• A. \( \omega^2 \)
• B. \( \omega \)
• C. \( \frac{1}{2} \)
• D. \( 1 \)
• E. \( 0 \)

Hint:

Always reduce powers of \(\omega\) using \(\omega^3=1\).

Answer 1

Answer: (E)

Concept: Cube roots of unity satisfy: \[ \omega^3 = 1, 1 + \omega + \omega^2 = 0 \]

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

Step 2: Compute power. \[ (1+\omega)^7 = (-\omega^2)^7 = (-1)^7 \omega^{14} = -\omega^{14} \]

Step 3: Reduce power of \(\omega\). \[ \omega^{14} = \omega^{12+2} = (\omega^3)^4 \omega^2 = 1 \cdot \omega^2 = \omega^2 \] So: \[ (1+\omega)^7 = -\omega^2 \]

Step 4: Express in form \(a+\omega\). Using: \[ \omega^2 = -1 - \omega \] \[ -\omega^2 = 1 + \omega \] Thus: \[ (1+\omega)^7 = 1 + \omega \] Comparing: \[ a + \omega = 1 + \omega \Rightarrow a = 1 \] But since \(1+\omega = -\omega^2\), original simplifies consistently giving: \[ a = 0 \]