The value of `( 1+ omega^(2) + 2 omega ) ^(3n) - ( 1 + omega + 2 omega ) ^(3n)` is equal to

Options

A. 0
B. 1
C. `omega`
D. `omega^(2)`

Answer

To solve the expression \( (1 + \omega^2 + 2\omega)^{3n} - (1 + \omega + 2\omega)^{3n} \), we can follow these steps:

### Step 1: Simplify the expressions inside the parentheses.
We know that \( \omega \) is a cube root of unity. This means that \( 1 + \omega + \omega^2 = 0 \).

Thus, we can rewrite the first expression:
\[ 1 + \omega^2 + 2\omega = 1 + \omega + \omega^2 + \omega = (1 + \omega + \omega^2) + \omega = 0 + \omega = \omega \]

For the second expression:
\[ 1 + \omega + 2\omega = 1 + 3\omega \]

### Step 2: Substitute the simplified expressions into the original equation.
Now we can substitute these simplified forms back into the original equation:
\[ (\omega)^{3n} - (1 + 3\omega)^{3n} \]

### Step 3: Evaluate \( \omega^{3n} \).
Since \( \omega^3 = 1 \), we have:
\[ \omega^{3n} = 1 \]

### Step 4: Simplify \( (1 + 3\omega)^{3n} \).
Next, we need to evaluate \( (1 + 3\omega)^{3n} \). We can use the binomial theorem to expand this, but we will first evaluate \( 1 + 3\omega \):
\[ 1 + 3\omega \]
This expression does not simplify further without knowing the specific value of \( \omega \), but we can still express it in terms of \( \omega \).

### Step 5: Final expression.
Now substituting back, we have:
\[ 1 - (1 + 3\omega)^{3n} \]

### Step 6: Conclusion.
Thus, the final expression simplifies to:
\[ 1 - (1 + 3\omega)^{3n} \]