If 1,`omega,omega^2,omega^3,...,omega^(n-1)` be the n,nth roots of unity,the value of `(9-omega)(9-omega^2)...(9-omega^(n-1))` will be :

Options

A. 0
B. `(9^(n-1))/8`
C. `(8^(n-1))/3`
D. none of these

Answer

To solve the problem, we need to find the value of the product \((9 - \omega)(9 - \omega^2) \cdots (9 - \omega^{n-1})\), where \(\omega\) is a primitive \(n\)th root of unity.

### Step-by-step Solution:

1. **Understanding Roots of Unity**:
The \(n\)th roots of unity are given by \(1, \omega, \omega^2, \ldots, \omega^{n-1}\), where \(\omega = e^{2\pi i/n}\). These roots satisfy the equation \(x^n - 1 = 0\).

2. **Factorization of Polynomial**:
The polynomial \(x^n - 1\) can be factored as:
\[ x^n - 1 = (x - 1)(x - \omega)(x - \omega^2) \cdots (x - \omega^{n-1}). \]

3. **Setting Up the Equation**:
We can express \(x^n - 1\) in terms of \(x\):
\[ x^n - 1 = 0 \implies x^n = 1. \]
Thus, we can write:
\[ x^n - 1 = (x - 1)(x - \omega)(x - \omega^2) \cdots (x - \omega^{n-1}). \]

4. **Substituting \(x = 9\)**:
We substitute \(x = 9\) into the equation:
\[ 9^n - 1 = (9 - 1)(9 - \omega)(9 - \omega^2) \cdots (9 - \omega^{n-1}). \]

5. **Calculating the Left Side**:
The left side becomes:
\[ 9^n - 1 = 9^n - 1. \]

6. **Calculating the Right Side**:
The right side simplifies to:
\[ 8(9 - \omega)(9 - \omega^2) \cdots (9 - \omega^{n-1}). \]

7. **Equating Both Sides**:
Now we equate both sides:
\[ 9^n - 1 = 8(9 - \omega)(9 - \omega^2) \cdots (9 - \omega^{n-1}). \]

8. **Isolating the Product**:
To isolate the product, we divide both sides by 8:
\[ \frac{9^n - 1}{8} = (9 - \omega)(9 - \omega^2) \cdots (9 - \omega^{n-1}). \]

9. **Final Result**:
Thus, the value of the product \((9 - \omega)(9 - \omega^2) \cdots (9 - \omega^{n-1})\) is:
\[ \frac{9^n - 1}{8}. \]