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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. \[{9^n} + 1\]

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Last updated date: 20th Jun 2024
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Answer
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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 ${\omega ^3} = 1$.