Question

If $\alpha$ and $\beta$ be two roots of the equation $x^2-64x+256=0.$ Then the value of ${\left(\frac{{\alpha }^3}{{\beta }^5}\right)}^{\frac{1}{8}}+{\left(\frac{{\beta }^3}{{\alpha }^5}\right)}^{\frac{1}{8}}$ is:

• A. $1$
• B. $3$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

$x^2-64x+256=0$
$\alpha +\beta =64$
$\alpha \beta =256$
${\left(\frac{{\alpha }^3}{{\beta }^5}\right)}^{\frac{1}{8}}+{\left(\frac{{\beta }^3}{{\alpha }^5}\right)}^{\frac{1}{8}}$
$=\frac{\alpha +\beta }{(\alpha \beta {)}^{\frac{5}{8}}}=\frac{64}{(256{)}^{\frac{5}{8}}}=\frac{64}{32}=2$