Question

rationalize √x+√y÷√x-√y​

Answer 1

Answer:

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Answer 2

Answer :-

$\bf \dfrac{x + y + 2 \sqrt{xy} }{x - y}$

Solution :-

$\sf \dfrac{ \sqrt{x} + \sqrt{y} }{ \sqrt{x} - \sqrt{y} }$

The rationalising factor of √x - √y is √x + √y. So, multiply both numerator and denominator by rationalising factor

$\sf = \dfrac{ \sqrt{x} + \sqrt{y} }{ \sqrt{x} - \sqrt{y}} \times \dfrac{ \sqrt{x} + \sqrt{y} }{ \sqrt{x} + \sqrt{y}}$

$\sf = \dfrac{ (\sqrt{x} + \sqrt{y})^2}{ (\sqrt{x})^{2} - (\sqrt{y})^{2} }$

[Since (a + b)(a - b) = a² - b² ]

$\sf = \dfrac{ (\sqrt{x} + \sqrt{y})^2}{x - y}$

$\sf = \dfrac{ (\sqrt{x})^2 + {( \sqrt{y})}^{2} + 2( \sqrt{x})( \sqrt{y})}{x - y}$

[Since (a + b)² = a² + b² + 2ab]

$\sf = \dfrac{x + y + 2 \sqrt{x \times y} }{x - y}$

$\sf = \dfrac{x + y + 2 \sqrt{xy} }{x - y}$

Identities used :-

• (a + b)(a - b) = a² - b²

• (a + b)² = a² + b² + 2ab