Question

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

Answer 1

EXPLANATION.

$\sf : \implies \: \alpha \: and \: \beta \: are \: roots \: of \: equation \: \implies \: 8 {x}^{2} - 3x + 27 = 0 \\ \\ \sf : \implies \: \: to \: find \: the \: value \: of \: (\dfrac{ { \alpha }^{2} }{ \beta } ) {}^{ \dfrac{1}{3} } \: + \: (\frac{ \beta {}^{2} }{ \alpha } ) {}^{ \dfrac{1}{3} }$

$\sf : \implies \: \: sum \: of \: roots \: of \: quadratic \: equation \\ \\ \sf : \implies \: \: \alpha + \beta = \frac{ - b}{a} \\ \\ \sf : \implies \: \: \alpha + \beta = \frac{3}{8} \: \: .......(1) \\ \\ \sf : \implies \: \: product \: of \: roots \: of \: quadratic \: equation \\ \\ \sf : \implies \: \: \alpha \beta = \frac{c}{a} \\ \\ \sf : \implies \: \: \alpha \beta = \frac{27}{8} \: \: ......(2)$

$\sf : \implies \: \: (\dfrac{ { \alpha }^{2} }{ \beta }) {}^{ \dfrac{1}{3} } \: + \: (\dfrac{ { \beta }^{2} }{ \alpha }) {}^{ \dfrac{1}{3} } \\ \\ \sf : \implies \: \: \frac{( { \alpha }^{3}) {}^{ \dfrac{1}{3} } + \: ( { \beta }^{3}) {}^{ \dfrac{1}{3} } }{( \alpha \beta ) {}^{ \dfrac{1}{3} } } \\ \\ \sf : \implies \: \: \frac{ \alpha + \beta }{( \alpha \beta ) {}^{ \dfrac{1}{3} } }$

$\sf : \implies \: \: \dfrac{ \dfrac{3}{8} }{( \frac{27}{8}) {}^{ \dfrac{1}{3} } } \\ \\ \sf : \implies \: \: \frac{ \frac{3}{8} }{( \frac{3}{2}) \dfrac{3}{3} } \implies \: \frac{1}{4} \: = \: answer$

Answer 2

$\sf \underline{ \underline{\pink{ \mathfrak{Solution}}}}$


$\sf \: {8x}^{2} - 3x + 27 = 0$



$\sf \: so \: R = \alpha + \beta = \frac{3}{8}$



$\sf \: POR = \alpha \beta = \frac{27}{8}$



$\large \sf \bigg( {\frac{ { \alpha}^{2} }{ \beta} \bigg)}^ { \frac{1}{3} } + \bigg( { \frac{ { \beta}^{2} }{ \alpha} \bigg) }^{ \frac{1}{3} } = \frac{ { \alpha}^{ \frac{1}{3} } }{ \beta \frac{ 1}{3} } \: + \: \frac{ { \beta}^{ \frac{2}{3} } }{ { \alpha}^{ \frac{1}{3} } }$



$\red{ \sf \large \implies} \frac{ { \alpha}^{ \frac{2}{3} } { \alpha}^{ \frac{1}{3} } + { \beta}^{ \frac{2}{3} } { \beta}^{ \frac{1}{3} } }{ { \alpha}^{ \frac{1}{3} } \: \: { \beta}^{ \frac{1}{3} } }$



$\red{ \sf \large \implies} \: \frac{ \alpha + \beta}{( \alpha \: \beta {)}^{ \frac{1}{3} } } = \frac{ \frac{3}{8} }{( \frac{27}{8} {)}^{3} }$




$\red{ \implies }\sf \large \: \frac{ \frac{3}{8} }{ \frac{3}{2} } = \frac{1}{4}$