The value of the expression (ω−1)ω−ω2ω−ω3…ω−ωn−1, where w is the nth root of unity, is

We have,xn−1=(x−1)(x−w)x−w2…x−wn−1⇒ xn−1x−w=(x−1)x−w2…x−wn−1Putting x = w on both sides, we have(w−1)w−w2…w−wn−1=limx→w xn−1x−w00=limx→w nxn−11=nwn−1

The value of the expression (ω−1)ω−ω2ω−ω3…ω−ωn−1, where w is the nth root of unity, is

We have,xn−1=(x−1)(x−w)x−w2…x−wn−1⇒ xn−1x−w=(x−1)x−w2…x−wn−1Putting x = w on both sides, we have(w−1)w−w2…w−wn−1=limx→w xn−1x−w00=limx→w nxn−11=nwn−1

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The value of the expression $(ω - 1) (ω - ω^{2}) (ω - ω^{3}) \dots (ω - ω^{n - 1})$ , where w is the n^th root of unity, is

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${nω}^{n - 1}$

$(n - 1) ω^{n}$

$(n - 1) ω^{n - 1}$

${nω}^{n}$

answer is A.

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Detailed Solution

We have,

$\begin{array}{l} x^{n} - 1 = (x - 1) (x - w) (x - w^{2}) \dots (x - w^{n - 1}) \\ \Rightarrow \frac{x^{n} - 1}{x - w} = (x - 1) (x - w^{2}) \dots (x - w^{n - 1}) \end{array}$

Putting x = w on both sides, we have

$\begin{matrix} (w - 1) (w - w^{2}) \dots (w - w^{n - 1}) = lim_{x \to w} \frac{x^{n} - 1}{x - w} (\frac{0}{0}) \\ = lim_{x \to w} \frac{{nx}^{n - 1}}{1} = {nw}^{n - 1} \end{matrix}$