Question

If $\omega$ is a complex cube root of unity, then $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 =$

• A. $32$
• B. $-32$
• C. $64$
• D. $-64$

Hint:

Always look to substitute $1 + \omega = -\omega^2$ and $1 + \omega^2 = -\omega$ when working with powers of complex cube roots of unity.

Answer 1

The Correct Option is A

Step 1: Concept
We use the properties of the complex cube roots of unity:

• $1 + \omega + \omega^2 = 0$

• $\omega^3 = 1$

Step 2: Meaning
We can rewrite the expressions inside the parentheses using $1 + \omega^2 = -\omega$ and $1 + \omega = -\omega^2$.

Step 3: Analysis
For the first term: \[ 1 - \omega + \omega^2 = (1 + \omega^2) - \omega = -\omega - \omega = -2\omega \] \[ (-2\omega)^5 = -32\omega^5 = -32\omega^2 \quad (\text{since } \omega^5 = \omega^3 \cdot \omega^2 = \omega^2) \] For the second term: \[ 1 + \omega - \omega^2 = (1 + \omega) - \omega^2 = -\omega^2 - \omega^2 = -2\omega^2 \] \[ (-2\omega^2)^5 = -32\omega^{10} = -32\omega \quad (\text{since } \omega^{10} = (\omega^3)^3 \cdot \omega = \omega) \] Adding the two terms together: \[ (-32\omega^2) + (-32\omega) = -32(\omega^2 + \omega) \] Using $\omega^2 + \omega = -1$: \[ -32(-1) = 32 \]

Step 4: Conclusion
The value of the complex expression is $32$.

Final Answer: (A)