The value of (1−ω+ω^2 )^ 5 +(1+ω−ω^ 2 )^ 5 where ω and ω^ 2 are the complex cube roots of unity, is

Options

A. 32
B. 16
C. 8
D. 0

Answer

To solve the problem, we need to evaluate the expression \( (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 \), where \( \omega \) and \( \omega^2 \) are the complex cube roots of unity.

### Step 1: Understand the properties of cube roots of unity
The complex cube roots of unity satisfy the equation:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can derive:
\[ \omega + \omega^2 = -1 \]
\[ \omega^2 = -1 - \omega \]
\[ \omega = -1 - \omega^2 \]

### Step 2: Rewrite the expressions
We will rewrite the terms \( (1 - \omega + \omega^2) \) and \( (1 + \omega - \omega^2) \) using the properties of \( \omega \).

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

2. For \( 1 + \omega - \omega^2 \):
\[ 1 + \omega - \omega^2 = 1 + \omega - (-1 - \omega) = 1 + \omega + 1 + \omega = 2 + 2\omega = 2(1 + \omega) \]
Since \( 1 + \omega = -\omega^2 \):
\[ 1 + \omega - \omega^2 = 2(-\omega^2) = -2\omega^2 \]

### Step 3: Substitute back into the expression
Now we substitute back into the original expression:
\[ (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 = (-2\omega)^5 + (-2\omega^2)^5 \]
Calculating each term:
\[ (-2\omega)^5 = -32\omega^5 \]
\[ (-2\omega^2)^5 = -32(\omega^2)^5 \]

### Step 4: Simplify \( \omega^5 \) and \( (\omega^2)^5 \)
Using the property \( \omega^3 = 1 \):
\[ \omega^5 = \omega^{3+2} = \omega^2 \]
\[ (\omega^2)^5 = (\omega^2)^{3+2} = \omega \]
Thus, we have:
\[ -32\omega^5 = -32\omega^2 \]
\[ -32(\omega^2)^5 = -32\omega \]

### Step 5: Combine the results
Now we combine the results:
\[ -32\omega^2 - 32\omega = -32(\omega + \omega^2) \]
Using \( \omega + \omega^2 = -1 \):
\[ -32(-1) = 32 \]

### Final Answer
The value of \( (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 \) is:
\[ \boxed{32} \]