Question

Please solve with full explaination​

Answer 1

⭐️ SOLUTION :-

$to \: find \: {x}^{3} - 3 {b}^{ \frac{2}{3} } x + 9a$

Given...

$x = (2a + \sqrt{4 {a}^{2} - {b}^{2} } )^{ \frac{1}{3} } + (2a - \sqrt{4 {a}^{2} - {b}^{2} } ) ^{ \frac{1}{3} }$

Now...

${x}^{3} = [(2a + \sqrt{4 {a}^{2} - {b}^{2} } )^{ \frac{1}{3} } + (2a - \sqrt{4 {a}^{2} - {b}^{2} } ) ^{ \frac{1}{3} } ]^{3}$

${x}^{3} = (2a + \sqrt{4 {a}^{2} - {b}^{2} } )^{ \frac{1}{3} \times 3 } + (2a - \sqrt{4 {a}^{2} - {b}^{2} } ) ^{ \frac{1}{3} \times 3} + 3 (2a + \sqrt{4 {a}^{2} - {b}^{2} } )^{ \frac{1}{3} } (2a - \sqrt{4 {a}^{2} - {b}^{2} } ) ^{ \frac{1}{3} } [(2a + \sqrt{4 {a}^{2} - {b}^{2} } )^{ \frac{1}{3} } + (2a - \sqrt{4 {a}^{2} - {b}^{2} } ) ^{ \frac{1}{3} }]$

${x}^{3} = (2a + \sqrt{4 {a}^{2} - {b}^{2} } ) + (2a - \sqrt{4 {a}^{2} - {b}^{2} } ) + 3 ( {(2a)}^{2} - ( \sqrt{4 {a}^{2} - {b}^{2} } ) ^{2} )^{ \frac{1}{3} } \times x$

${x}^{3} = 2a + \sqrt{4 {a}^{2} - {b}^{2} } + 2a - \sqrt{4 {a}^{2} - {b}^{2} } + 3 ( 4{a}^{2} - 4 {a}^{2} + {b}^{2} )^{ \frac{1}{3} } \times x$

${x}^{3} = 2a + 2a + 3 {b}^{ \frac{2}{3} } x$

${x}^{3} = 4a + 3 {b}^{ \frac{2}{3} } x$

Therefore, putting the values in the equation...

${x}^{3} - 3 {b}^{ \frac{2}{3} } x + 9a = 4a + 3 {b}^{ \frac{2}{3} } x - 3 {b}^{ \frac{2}{3} } x + 9a$

${x}^{3} - 3 {b}^{ \frac{2}{3} } x + 9a = 4a + 9a$

${x}^{3} - 3 {b}^{ \frac{2}{3} } x + 9a =13a$

So, option (2) 13a is correct ✔️...

HOPE IT HELPS ☺