Question

if $\alpha and\beta$ are complex cube root of unity then find the value of ${\alpha }^2+{\beta }^2+\alpha \beta$

Answer 1

Complex cube roots are $\alpha =\frac{-1+i\sqrt{3}}{2}$ and $\beta =\frac{-1-i\sqrt{3}}{2}$

$\Rightarrow \alpha +\beta =\frac{-1+i\sqrt{3}}{2}+\frac{-1-i\sqrt{3}}{2}=-1$

$\Rightarrow \alpha \beta =\frac{-1+i\sqrt{3}}{2}\times \frac{-1-i\sqrt{3}}{2}=\frac{{(-1)}^2-{\left(i\sqrt{3}\right)}^2}{4}=\frac{1+3}{4}=1$

Now, ${\alpha }^2+{\beta }^2+\alpha \beta$

$={(\alpha +\beta )}^2-2\alpha \beta +\alpha \beta$

$={(\alpha +\beta )}^2-\alpha \beta$

$={(-1)}^2-1=1-1=0$