Question

If $\alpha ,\beta$ are the complex cube roots of unity then ${\alpha }^{100}+{\beta }^{100}+\frac{1}{{\alpha }^{100}\times {\beta }^{100}}=?$

Answer 1

let $\alpha =\omega$ & $\beta ={\omega }^2$

$\therefore {\alpha }^{100}+{\beta }^{100}+\frac{1}{{\alpha }^{100}\times {\beta }^{100}}={\omega }^{100}+{\omega }^{200}+\frac{1}{{\omega }^{100}\times {\omega }^{200}}$

${\alpha }^{100}+{\beta }^{100}+\frac{1}{{\alpha }^{100}\times {\beta }^{200}}={\omega }^{100}+{\omega }^{200}+\frac{1}{{\omega }^{300}}$

${\alpha }^{100}+{\beta }^{100}+\frac{1}{{\omega }^{100}\times {\omega }^{200}}=\omega +{\omega }^2+\frac{1}{1}$

$({\omega }^{3n}=1,{\omega }^{3n+1}=\omega \&{\omega }^{3n+2}={\omega }^2)$

${\alpha }^{100}+{\beta }^{100}+\frac{1}{{\alpha }^{100}\times {\beta }^{200}}=(-1)+1\leftarrow (1+\omega +{\omega }^2=0)$

${\alpha }^{100}+{\beta }^{100}+\frac{1}{{\alpha }^{100}\ast {\beta }^{200}}=0$