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: Understanding the Concept
For a complex cube root of unity, \(1+\omega+\omega^{2}=0\) and \(\omega^{3}=1\).

Step 2: Simplify the Brackets
\(1-\omega+\omega^{2}=-\omega-\omega=-2\omega\).
\(1+\omega-\omega^{2}=-\omega^{2}-\omega^{2}=-2\omega^{2}\).

Step 3: Raise to the Power 7
\((-2\omega)^{7}=-128\omega^{7}=-128\omega\).
\((-2\omega^{2})^{7}=-128\omega^{14}=-128\omega^{2}\).

Step 4: Add
\(-128(\omega+\omega^{2})=-128(-1)=128\).

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