Question

If \( \omega \) is an imaginary cube root of unity, find the value of \( (1 + \omega - \omega^2)^7 \).

• A. \(128\omega^2\)
• B. \(-128\omega^2\)
• C. \(64\omega\)
• D. \(-64\omega^2\)

Hint:

For cube roots of unity always remember: \[ 1+\omega+\omega^2=0,\qquad \omega^3=1 \] Reduce higher powers using \( \omega^3=1 \) and simplify expressions quickly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks to evaluate an algebraic expression involving the complex cube root of unity \( \omega \).
Step 2: Key Formula or Approach:
The fundamental properties of cube roots of unity are:
1. \( 1 + \omega + \omega^2 = 0 \) (Sum of roots)
2. \( \omega^3 = 1 \) (Product of roots)
Step 3: Detailed Explanation:
From the property \( 1 + \omega + \omega^2 = 0 \), we can derive:
\[ 1 + \omega = -\omega^2 \]
Substitute this into the given expression \( (1 + \omega - \omega^2)^7 \):
\[ ( (-\omega^2) - \omega^2 )^7 = (-2\omega^2)^7 \]
Expand the power:
\[ (-2)^7 \cdot (\omega^2)^7 = -128 \cdot \omega^{14} \]
Now, simplify \( \omega^{14} \) using \( \omega^3 = 1 \):
\[ \omega^{14} = \omega^{12} \cdot \omega^2 = (\omega^3)^4 \cdot \omega^2 \]
\[ \omega^{14} = (1)^4 \cdot \omega^2 = \omega^2 \]
Substituting this back:
\[ -128\omega^{14} = -128\omega^2 \]
Step 4: Final Answer:
The value of the expression is \( -128\omega^2 \).