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# If $1,\omega ,\;{\omega ^2},...............{\omega ^{n - 1}}$ are $n,{n^{th}}$ roots of unity, then the value of $\left( {9 - \omega } \right)\left( {9 - {\omega ^2}} \right)\left( {9 - {\omega ^3}} \right)...............\left( {9 - {\omega ^{n - 1}}} \right)$ will be A. $\dfrac{{{9^n} + 1}}{8}$B. ${9^n} - 1$C. $\dfrac{{{9^n} - 1}}{8}$D. ${9^n} + 1$

Last updated date: 20th Jun 2024
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Hint: In this question, we will proceed by taking $x = {\left( 1 \right)^{\dfrac{1}{n}}}$ and then raising on both sides by $n$. Then convert this into an equation and apply binomial expansion. Further substitute $x = 9$ to get the required answer.

Complete step-by-step answer:
Here we have to find the value of given expression $\left( {9 - \omega } \right)\left( {9 - {\omega ^2}} \right)\left( {9 - {\omega ^3}} \right)...............\left( {9 - {\omega ^{n - 1}}} \right)$
Let’s say $x = {\left( 1 \right)^{\dfrac{1}{n}}}$
And hence on taking power $n$ on both sides, we have
$\Rightarrow {x^n} = {\left( {{1^{\dfrac{1}{n}}}} \right)^n} \\ \Rightarrow {x^n} = 1 \\ \Rightarrow {x^n} - 1 = 0 \\$
We know that ${x^n} - 1 = \left( {x - 1} \right)\left( {x - \omega } \right)\left( {x - {\omega ^2}} \right)...............\left( {x - {\omega ^{n - 1}}} \right)$
Dividing with $x - 1$ on both sides, we have
$\Rightarrow \dfrac{{{x^n} - 1}}{{x - 1}} = \left( {x - \omega } \right)\left( {x - {\omega ^2}} \right)...............\left( {x - {\omega ^{n - 1}}} \right)$
Put $x = 9$, then we have
$\Rightarrow \dfrac{{{9^n} - 1}}{{9 - 1}} = \left( {9 - \omega } \right)\left( {9 - {\omega ^2}} \right)..................\left( {9 - {\omega ^{n - 1}}} \right) \\ \Rightarrow \dfrac{{{9^n} - 1}}{8} = \left( {9 - \omega } \right)\left( {9 - {\omega ^2}} \right)..................\left( {9 - {\omega ^{n - 1}}} \right) \\$
And hence the value of $\left( {9 - \omega } \right).\left( {9 - {\omega ^2}} \right).\left( {9 - {\omega ^3}} \right)...............\left( {9 - {\omega ^{n - 1}}} \right)$ is equals to
$\Rightarrow \dfrac{{{9^n} - 1}}{8}$
Thus, the correct option is C. $\dfrac{{{9^n} - 1}}{8}$

So, the correct answer is “Option C”.

Note: Here we have used the binomial expansion ${x^n} - 1 = \left( {x - 1} \right)\left( {x - \omega } \right)\left( {x - {\omega ^2}} \right)...............\left( {x - {\omega ^{n - 1}}} \right)$. Always remember that $1 + \omega + {\omega ^2} = 0$ and ${\omega ^3} = 1$.