If \(\omega\) is an imaginary cube root of unity, then the value of \(\left(1-\omega+\omega^{2}\right) \cdot\left(1-\omega^{2}+\omega^{4}\right) \cdot\left(1-\omega^{4}+\omega^{8}\right) \cdot \ldots\)
(2n factors) is

Given, \(\quad \omega^{3}=1,1+\omega+\omega^{2}=0\)
Now,
\(\left(1-\omega+\omega^{2}\right) \cdot\left(1-\omega^{2}+\omega^{4}\right) \cdot\left(1-\omega^{4}+\omega^{8}\right)\).
\(\left(1-\omega^{8}+\omega^{16}\right) \ldots 2 n\) factors \(=\left(1-\omega+\omega^{2}\right)\left(1-\omega^{2}+\omega\right)\)
\(\cdot\left(1-\omega+\omega^{2}\right) \cdot\left(1-\omega^{2}+\omega\right) \ldots .2 n\) factors
[from Eq. (i)]
\(=(-\omega-\omega) \cdot\left(-\omega^{2}-\omega^{2}\right) \cdot(-\omega-\omega) \cdot\left(-\omega^{2}-\omega^{2}\right)\)
... \(2 n\) factors [from Eq. (i)]
\(=(-2 \omega) \cdot\left(-2 \omega^{2}\right) \cdot(-2 \omega) \cdot\left(-2 \omega^{2}\right) \ldots 2 n\) factors \(=(-2)^{2 n}(\omega)^{n}\left(\omega^{2}\right)^{n}\) \(=(-1)^{2 n}(2)^{2 n} \omega^{3 n}\) \(=2^{2 n} \cdot\left(\omega^{3}\right)^{n}=2^{2 n}(1)^{n}=2^{2 n} \quad\) [from Eq. (i)]