Evaluate : `(2-omega^(100))(2-omega^(101))(2-omega^(10))(2-omega^(11))`

Answer

To evaluate the expression \((2 - \omega^{100})(2 - \omega^{101})(2 - \omega^{10})(2 - \omega^{11})\), we will use the properties of \(\omega\), where \(\omega\) is a primitive cube root of unity. We know that:

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

### Step 1: Simplifying \(\omega^{100}\) and \(\omega^{101}\)

First, we simplify \(\omega^{100}\) and \(\omega^{101}\):
- \(\omega^{100} = \omega^{3 \cdot 33 + 1} = (\omega^3)^{33} \cdot \omega^1 = 1^{33} \cdot \omega = \omega\)
- \(\omega^{101} = \omega^{3 \cdot 33 + 2} = (\omega^3)^{33} \cdot \omega^2 = 1^{33} \cdot \omega^2 = \omega^2\)

### Step 2: Simplifying \(\omega^{10}\) and \(\omega^{11}\)

Next, we simplify \(\omega^{10}\) and \(\omega^{11}\):
- \(\omega^{10} = \omega^{3 \cdot 3 + 1} = (\omega^3)^3 \cdot \omega^1 = 1^3 \cdot \omega = \omega\)
- \(\omega^{11} = \omega^{3 \cdot 3 + 2} = (\omega^3)^3 \cdot \omega^2 = 1^3 \cdot \omega^2 = \omega^2\)

### Step 3: Substitute the values into the expression

Now we substitute these values back into the original expression:
\[ (2 - \omega)(2 - \omega^2)(2 - \omega)(2 - \omega^2) \]
This can be rewritten as:
\[ (2 - \omega)^2(2 - \omega^2)^2 \]

### Step 4: Calculate \((2 - \omega)(2 - \omega^2)\)

Next, we need to calculate \((2 - \omega)(2 - \omega^2)\):
\[ (2 - \omega)(2 - \omega^2) = 2^2 - 2(\omega + \omega^2) + \omega \cdot \omega^2 \]
Using the property \(\omega + \omega^2 = -1\) and \(\omega \cdot \omega^2 = \omega^3 = 1\):
\[ = 4 - 2(-1) + 1 = 4 + 2 + 1 = 7 \]

### Step 5: Final calculation

Now we substitute back into our expression:
\[ (2 - \omega)^2(2 - \omega^2)^2 = (7)^2 = 49 \]

### Final Answer

Thus, the value of the expression is:
\[ \boxed{49} \]