Let `omega(omega ne 1)` is a cube root of unity, such that `(1+omega^(2))^(8)=a+bomega` where a, b in R, then `|a+b|` is equal to

Options

A. 1
B. 3
C. 0
D. 2

Answer

To solve the problem, we need to find the value of \(|a + b|\) given that \((1 + \omega^2)^8 = a + b\omega\), where \(\omega\) is a cube root of unity and \(\omega \neq 1\).

### Step-by-Step Solution:

1. **Understanding Cube Roots of Unity**:
The cube roots of unity are the solutions to the equation \(x^3 = 1\). They are \(1\), \(\omega\), and \(\omega^2\), where \(\omega = e^{2\pi i / 3}\) and \(\omega^2 = e^{-2\pi i / 3}\). We know that:
\[ 1 + \omega + \omega^2 = 0 \implies \omega + \omega^2 = -1. \]

2. **Expressing \(1 + \omega^2\)**:
From the relation above, we can express \(1 + \omega^2\) as:
\[ 1 + \omega^2 = -\omega. \]

3. **Substituting into the Equation**:
Now, substitute \(1 + \omega^2 = -\omega\) into the equation:
\[ (1 + \omega^2)^8 = (-\omega)^8. \]

4. **Calculating \((- \omega)^8\)**:
Since \((- \omega)^8 = (-1)^8 \cdot \omega^8 = 1 \cdot \omega^8 = \omega^8\).

5. **Finding \(\omega^8\)**:
To find \(\omega^8\), we can reduce the exponent modulo 3 (since \(\omega^3 = 1\)):
\[ 8 \mod 3 = 2 \implies \omega^8 = \omega^2. \]

6. **Setting the Equation**:
Now we have:
\[ \omega^2 = a + b\omega. \]

7. **Expressing \(\omega^2\)**:
We can express \(\omega^2\) in terms of \(a\) and \(b\):
\[ \omega^2 = -1 - \omega. \]

8. **Comparing Coefficients**:
Now we compare the coefficients of \(\omega\):
- The constant term gives us \(a = -1\).
- The coefficient of \(\omega\) gives us \(b = -1\).

9. **Calculating \(a + b\)**:
Now we can find \(a + b\):
\[ a + b = -1 - 1 = -2. \]

10. **Finding the Modulus**:
Finally, we need to find \(|a + b|\):
\[ |a + b| = |-2| = 2. \]

### Final Answer:
\[ |a + b| = 2. \]