`(3+omega + 3 omega^(2))^(4)= lambda omega`, then value of `lambda ` is :

Options

A. 4
B. `-4`
C. 16
D. `-8`

Answer

To solve the equation \((3 + \omega + 3\omega^2)^4 = \lambda \omega\), we will follow these steps:

### Step 1: Simplify the expression inside the parentheses
We start with the expression \(3 + \omega + 3\omega^2\). We can rewrite \(3\) as \(2 + 1\):

\[ 3 + \omega + 3\omega^2 = (2 + 1) + \omega + 3\omega^2 = 2 + (1 + \omega + 3\omega^2) \]

### Step 2: Use the property of cube roots of unity
We know that the cube roots of unity are \(1, \omega, \omega^2\) and they satisfy the equation \(1 + \omega + \omega^2 = 0\). Therefore:

\[ 1 + \omega + 3\omega^2 = 1 + \omega + \omega^2 + 2\omega^2 = 0 + 2\omega^2 = 2\omega^2 \]

So we can substitute this back into our expression:

\[ 3 + \omega + 3\omega^2 = 2 + 2\omega^2 = 2(1 + \omega^2) \]

### Step 3: Substitute back into the original expression
Now we can rewrite the original expression:

\[ (3 + \omega + 3\omega^2)^4 = (2(1 + \omega^2))^4 = 2^4(1 + \omega^2)^4 \]

### Step 4: Simplify \(1 + \omega^2\)
Using the property of cube roots of unity again, we have:

\[ 1 + \omega^2 = -\omega \]

### Step 5: Substitute and simplify further
Now substituting this back, we get:

\[ (3 + \omega + 3\omega^2)^4 = 2^4(-\omega)^4 \]

Since \((-1)^4 = 1\) and \(\omega^4 = \omega\), we have:

\[ = 2^4 \cdot \omega = 16\omega \]

### Step 6: Compare with \(\lambda \omega\)
Now we can compare this with the original equation:

\[ 16\omega = \lambda \omega \]

### Step 7: Solve for \(\lambda\)
Dividing both sides by \(\omega\) (assuming \(\omega \neq 0\)) gives:

\[ \lambda = 16 \]

Thus, the value of \(\lambda\) is:

\[ \boxed{16} \]