If `omega` is a complex cube root of unity then `(1-omega+omega^2)(1-omega^2+omega^4)(1-omega^4+omega^8)(1-omega^8+omega^16)`

Options

A. 12
B. 14
C. 16
D. None of these

Answer

To solve the problem, we need to evaluate the expression:

\[ (1 - \omega + \omega^2)(1 - \omega^2 + \omega^4)(1 - \omega^4 + \omega^8)(1 - \omega^8 + \omega^{16}) \]

where \(\omega\) is a complex cube root of unity. We know that:

1. \(\omega^3 = 1\)
2. \(1 + \omega + \omega^2 = 0\)

### Step 1: Simplifying Each Factor

**First Factor:**
\[ 1 - \omega + \omega^2 = 1 + \omega^2 - \omega = 1 + \omega^2 - \omega \]
Using \(1 + \omega + \omega^2 = 0\), we can rewrite this as:
\[ 1 - \omega + \omega^2 = -\omega \]

**Second Factor:**
\[ 1 - \omega^2 + \omega^4 \]
Since \(\omega^4 = \omega\) (because \(\omega^4 = \omega^{3+1} = \omega\)), we have:
\[ 1 - \omega^2 + \omega = 1 + \omega - \omega^2 \]
Again using \(1 + \omega + \omega^2 = 0\):
\[ 1 - \omega^2 + \omega = -\omega^2 \]

**Third Factor:**
\[ 1 - \omega^4 + \omega^8 \]
Since \(\omega^4 = \omega\) and \(\omega^8 = \omega^2\):
\[ 1 - \omega + \omega^2 = -\omega \]

**Fourth Factor:**
\[ 1 - \omega^8 + \omega^{16} \]
Since \(\omega^8 = \omega^2\) and \(\omega^{16} = \omega\):
\[ 1 - \omega^2 + \omega = -\omega^2 \]

### Step 2: Putting It All Together

Now substituting back into the original expression:
\[ (-\omega)(-\omega^2)(-\omega)(-\omega^2) \]
This simplifies to:
\[ \omega^2 \cdot \omega^4 = \omega^6 \]

### Step 3: Evaluating \(\omega^6\)

Since \(\omega^3 = 1\):
\[ \omega^6 = (\omega^3)^2 = 1^2 = 1 \]

### Final Result

Thus, the value of the expression is:
\[ \boxed{1} \]