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(B) Given that $1, \omega, \omega^2, \ldots, \omega^8$ are the roots of the equation $x^9-1=0$. Since $\omega$ is a $9^{th}$ root of unity,we have $\o

Vedclass · Accepted Answer

$8$

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(B) Given that $1, \omega, \omega^2, \ldots, \omega^8$ are the roots of the equation $x^9-1=0$. Since $\omega$ is a $9^{th}$ root of unity,we have $\omega^9 = 1$. We need to evaluate the sum $S = \sum_{r=1}^8 \left(\omega^r\right)^{99}$. $S = \sum_{r=1}^8 \omega^{99r} = \sum_{r=1}^8 (\omega^9)^{11r}$. Since $\omega^9 = 1$,we have $(\omega^9)^{11r} = (1)^{11r} = 1$. Thus,$S = \sum_{r=1}^8 1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8$. Therefore,the correct option is $B$.

If $1, \omega, \omega^2, \ldots, \omega^8$ are the roots of the equation $x^9-1=0$,then $\sum_{r=1}^8 \left(\omega^r\right)^{99} =$

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