If ` 1 , omega, omega^(2)` are cube roots of unity then the value of `(5+2omega +5omega^(2))^(3)` is

Options

A. 27
B. `-9`
C. `-27`
D. `-81`

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ (5 + 2\omega + 5\omega^2)^3 \]
We can factor out common terms. Notice that we can group \( 2\omega + 5\omega^2 \):
\[ = (5 + 2\omega + 5\omega^2)^3 = (5 + 2\omega + 5\omega^2)^3 \]

### Step 2: Use the property of cube roots of unity
We know that for cube roots of unity:
\[ 1 + \omega + \omega^2 = 0 \]
This implies:
\[ \omega^2 = -1 - \omega \]
Substituting \( \omega^2 \) in the expression gives:
\[ 5 + 2\omega + 5(-1 - \omega) = 5 + 2\omega - 5 - 5\omega = 2\omega - 5\omega = -3\omega \]

### Step 3: Substitute back into the expression
Now we can substitute back into our original expression:
\[ (5 + 2\omega + 5\omega^2)^3 = (-3\omega)^3 \]

### Step 4: Calculate the cube
Calculating the cube gives:
\[ (-3\omega)^3 = -27\omega^3 \]

### Step 5: Use the property of \( \omega^3 \)
Since \( \omega^3 = 1 \), we can replace \( \omega^3 \) with 1:
\[ -27\omega^3 = -27 \cdot 1 = -27 \]

### Final Answer
Thus, the value of \( (5 + 2\omega + 5\omega^2)^3 \) is:
\[ \boxed{-27} \]