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

We have,

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

Putting x = w on both sides, we have

$\begin{array}{c}(\mathrm{w}-1)\left(\mathrm{w}-{\mathrm{w}}^2\right)\dots \left(\mathrm{w}-{\mathrm{w}}^{\mathrm{n}-1}\right)=\underset{\mathrm{x}\to \mathrm{w}}{lim} \frac{{\mathrm{x}}^{\mathrm{n}}-1}{\mathrm{x}-\mathrm{w}}\left(\frac{0}{0}\right)\\ =\underset{\mathrm{x}\to \mathrm{w}}{lim} \frac{{\mathrm{nx}}^{\mathrm{n}-1}}{1}={\mathrm{nw}}^{\mathrm{n}-1}\end{array}$