If α and β be the roots of the equation   x 2 - 2 x + 2 = 0 , then the least value of n for…

$x^2-2x+2=0$

$\Rightarrow$ ${(x-1)}^2+1=0$

$\Rightarrow x=1\pm i , i=\sqrt{-1}$

Now let the roots $\alpha =i+i \& \beta =1-i$

${\left(\frac{\alpha }{\beta }\right)}^n={\left(\frac{1+i}{1-i}\right)}^n={\left(\frac{i\left(1-i\right)}{1-i}\right)}^n$

$\Rightarrow {\left(\frac{\alpha }{\beta }\right)}^n={\left(i\right)}^n$

Using given information,

$\Rightarrow {\left(i\right)}^n=1$

$\Rightarrow$ The minimum value of $n\in N=4$ because ${\left(i\right)}^4=1$.