If α and β be the roots of the equation x ^ 2 - 2x + 2 = 0, then...

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

$⇒$ ${( x - 1 )}^{2} + 1 = 0$

$⇒ x = 1 \pm i, i = \sqrt{- 1}$

Now let the roots $\alpha = i + i \text{ and } \beta = 1 - i$

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

$⇒ {\left(\dfrac{\alpha}{\beta}\right)}^{n} = {( i )}^{n}$

Using given information,

$⇒ {( i )}^{n} = 1$

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