Question

If \( 1, \omega, \omega^2, \ldots, \omega^{n-1} \) are \( n \)th roots of unity, then the value of \( (9-\omega)(9-\omega^2)\cdots(9-\omega^{n-1}) \) is

• A. $\frac{9^n + 1}{8}$
• B. $9^n - 1$
• C. $\frac{9^n - 1}{8}$
• D. $9^n + 1$

Hint:

Always use $x^n - 1$ identity for roots of unity products.

Answer 1

The Correct Option is C

Understanding the Concept: Product of $(x - \omega^k)$ identity

Step 1: \[ (x-1)(x-\omega)\cdots(x-\omega^{n-1}) = x^n - 1 \]

Step 2: Put $x = 9$: \[ (9-1)(9-\omega)\cdots(9-\omega^{n-1}) = 9^n - 1 \]

Step 3: Divide by $(9-1)=8$: \[ (9-\omega)(9-\omega^2)\cdots = \frac{9^n - 1}{8} \] Final Answer:
Answer = $\frac{9^n - 1}{8}$