If α and β be two roots of the equation x ^ 2 - 64x + 256 = 0 ....

Given equation is ${x}^{2} - 64x + 256 = 0$ whose roots are $\alpha, \beta$.

$\alpha + \beta = 64, \alpha\beta = 256$
Now \left(\dfrac{{\alpha}^{3}}{{\beta}^{5}}\right)\dfrac{1}{8} + \left(\dfrac{{\beta}^{3}}{{\alpha}^{5}}\right)\dfrac{1}{8}$<br />$= \dfrac{{\alpha}^{\dfrac{3}{8} + \dfrac{5}{8}} + {\beta}^{\dfrac{3}{8} + \dfrac{5}{8}}}{( \alpha\beta{)}^{5 / 8}} = \dfrac{\alpha + \beta}{( \alpha\beta{)}^{5 / 8}}$<br />$= \dfrac{64}{( 256{)}^{5 / 8}} = \dfrac{64}{32} = 2$