Question

prove that , √3-√2/√3+√2 + √3+√2/√3-√2 = 10​

Answer 1

To Prove :-

${ \sf { \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} + \dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} = 10 }}$

Solution :-

${ \bf Let \:\: LHS \:\: i.e \:\: \sf \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} + \dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} }$

Take LCM ;

${ : \implies \quad { \sf { \dfrac{{(\sqrt{3} - \sqrt{2})²} + {(\sqrt{3} + \sqrt{2})²}}{(\sqrt{3} - \sqrt{2}) ( \sqrt{3} + \sqrt{2} ) } }}}$

We knows the following algebraic identities ;

$\quad \qquad { \bigstar { \underline {\boxed { \red { \bf { ( a + b )² = a² + b² + 2ab }}}}}}{\bigstar}$

$\quad \qquad { \bigstar { \underline {\boxed { \green { \bf { ( a - b )² = a² + b² - 2ab }}}}}}{\bigstar}$

$\quad \qquad { \bigstar { \underline {\boxed { \blue { \bf { a² - b² = ( a + b ) ( a - b ) }}}}}}{\bigstar}$

Using these identities we have ;

${ : \implies \quad { \sf ( 29 )² = ( 20 )² + x² }}$

${ : \implies { \sf { \dfrac{\{(\sqrt{2})² + (\sqrt{3})² - 2 × \sqrt{3} × \sqrt{2}\} + \{(\sqrt{3})² + (\sqrt{2})² + 2× \sqrt{3} × \sqrt{2}\}}{(\sqrt{3})² - (\sqrt{2})²} }}}$

${ : \implies { \sf { \dfrac{2 + 3 - 2\sqrt{6} + 2 + 3 + 2\sqrt{6}}{3 - 2} }}}$

${ : \implies { \sf { \dfrac{2 + 3 - \cancel{2\sqrt{6}} + 2 + 3 + \cancel{2\sqrt{6}}}{1} }}}$

${ : \implies { \sf { 2 + 2 + 3 + 3 }}}$

${ : \implies { \bf { 10 }}}$

Hence Proved !

${ \bigstar { \underline { \boxed { \red { \bf { \therefore \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} + \dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} = 10 }}}}}}{\bigstar}$