If `1, omega, omega^(2)` are the three cube roots of unity, then simplify: `(3 + 5omega + 3omega^(2))^(2) (1 + 2omega + omega^(2))`

Answer

To simplify the expression \((3 + 5\omega + 3\omega^2)^2 (1 + 2\omega + \omega^2)\), where \(1, \omega, \omega^2\) are the cube roots of unity, we can follow these steps:

### Step 1: Use the property of cube roots of unity
Recall that the cube roots of unity satisfy the equation:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can express \(\omega^2\) in terms of \(\omega\):
\[ \omega^2 = -1 - \omega \]

### Step 2: Simplify the first part \((3 + 5\omega + 3\omega^2)^2\)
Substituting \(\omega^2\) into the expression:
\[ 3 + 5\omega + 3\omega^2 = 3 + 5\omega + 3(-1 - \omega) = 3 + 5\omega - 3 - 3\omega = 2\omega \]
Now we can square this:
\[ (2\omega)^2 = 4\omega^2 \]

### Step 3: Simplify the second part \((1 + 2\omega + \omega^2)\)
Using the same substitution for \(\omega^2\):
\[ 1 + 2\omega + \omega^2 = 1 + 2\omega + (-1 - \omega) = 2\omega \]

### Step 4: Combine the results
Now we can combine the results from Step 2 and Step 3:
\[ (3 + 5\omega + 3\omega^2)^2 (1 + 2\omega + \omega^2) = (4\omega^2)(2\omega) = 8\omega^3 \]

### Step 5: Use the property of cube roots of unity again
Since \(\omega^3 = 1\):
\[ 8\omega^3 = 8 \cdot 1 = 8 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{8} \]