Question

If $\alpha$ and $\beta$ are the roots of the equation $8x^2-3x+27=0,$ then the value of ${\left(\frac{{\alpha }^2}{\beta }\right)}^{1/3}+{\left(\frac{{\beta }^2}{\alpha }\right)}^{1/3}$ is

• A. $\frac{1}{6}$
• B. $\frac{1}{3}$
• C. $\frac{1}{5}$
• D. $\frac{1}{4}$

Answer 1

The correct option is D $\frac{1}{4}$

Give that $\alpha$ and $\beta$ are the roots of the equation $8x^2-3x+27=0.$
$\therefore \alpha +\beta =\frac{3}{8}$ and $\alpha \beta =\frac{27}{8}$

Now, ${\left(\frac{{\alpha }^2}{\beta }\right)}^{1/3}+{\left(\frac{{\beta }^2}{\alpha }\right)}^{1/3}$
$=\frac{({\alpha }^3{)}^{1/3}+({\beta }^3{)}^{1/3}}{(\alpha \beta {)}^{1/3}}=\frac{3/8}{3/2}=\frac{1}{4}$