Question

If \(\omega\) is a complex cube root of unity, then the value of the expression \((1-\omega +\omega ^{2})^{7}+(1+\omega -\omega ^{2})^{7}\) is equal to...

• A. \(128\)
• B. \(-128\)
• C. \(128\omega\)
• D. \(0\)

Hint:

Use 1+w+w^2=0 to rewrite each bracket as a multiple of w or w squared.

Answer 1

The Correct Option is A

Step 1: Replace using the Identity
$1+\omega^{2}=-\omega$, so the first bracket is $-2\omega$; similarly $1+\omega=-\omega^{2}$, so the second bracket is $-2\omega^{2}$.

Step 2: Reduce Powers
$\omega^{7}=\omega$ and $\omega^{14}=\omega^{2}$.

Step 3: Sum
The total is $-128(\omega+\omega^{2})=128$, option (A).

Final Answer:
The expression equals 128, option (A). \[ \boxed{\text{(A) }128} \]