Question

pls solve the question given below.

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:\dfrac{ \sqrt{13} - \sqrt{11} }{ \sqrt{13} + \sqrt{11} } = a - b \sqrt{143}$

On rationalizing the denominator, we get

$\rm :\longmapsto\:\dfrac{ \sqrt{13} - \sqrt{11} }{ \sqrt{13} + \sqrt{11} } \times \dfrac{ \sqrt{13} - \sqrt{11} }{ \sqrt{13} - \sqrt{11} } = a - b \sqrt{143}$

$\rm :\longmapsto\:\dfrac{( \sqrt{13} - \sqrt{11})^{2} }{( \sqrt{13})^{2} - (\sqrt{11})^{2} } = a - b \sqrt{143}$

$\red{\bigg \{ \because \: (x + y)(x - y) = {x}^{2} - {y}^{2} \bigg \}}$

$\rm :\longmapsto\: \dfrac{13 + 11 - 2 \sqrt{143} }{13 - 11} = a - b \sqrt{143}$

$\red{\bigg \{ \because \: {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy\bigg \}}$

$\rm :\longmapsto\: \dfrac{24 - 2 \sqrt{143} }{2} = a - b \sqrt{143}$

$\rm :\longmapsto\: \dfrac{2(12 - \sqrt{143} )}{2} = a - b \sqrt{143}$

$\rm :\longmapsto\: 12 - \red{\sqrt{143}} = a - \red{ b \sqrt{143}}$

So, on comparing we get

$\bf\implies \:a = 12 \: \: \: \: and \: \: \: \: b = 1$

Additional Information :-

More Identities to know:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)