Question

: Rationalise the denominator of
√2+√5/√2-√5​

Answer 1

$\huge{\mathcal{\underline{\underline{\blue{ Question : }}}}}$

Rationalise the denominator of √2 + √5/√2 - √5

$\huge{\mathcal{\underline{\underline{\blue{ Solution : }}}}}$

★ Given that,

Rationalise the denominator of √2 + √5/√2 - √5

★ Let,

$\rm\:\implies \frac{\sqrt{2} + \sqrt{5} }{\sqrt{2} - \sqrt{5}}$

• Rationalise the denominator with √2 + √5

$\rm\:\implies \frac{\sqrt{2} + \sqrt{5} }{\sqrt{2} - \sqrt{5}} \times \frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} + \sqrt{5}}$

$\rm\:\implies \frac{(\sqrt{2} + \sqrt{5})^{2}}{(\sqrt{2} - \sqrt{5})(\sqrt{2} + \sqrt{5})}$

• (a + b)² = a² + b² + 2ab
• (a + b)(a - b) = a² - b²

$\rm\:\implies \frac{(\sqrt{2})^{2} + (\sqrt{5})^{2} + 2(\sqrt{2})(\sqrt{5})}{(\sqrt{2})^{2} - (\sqrt{5})^{2}}$

$\rm\:\implies \frac{2 + 5 + 2\sqrt{10}}{2 - 5}$

$\rm\:\implies \frac{7 + 2\sqrt{10}}{-3}$

$\underline{\boxed{\bf{\purple{ \therefore \frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} - \sqrt{2} - \sqrt{5}} = - \frac{7 + 2\sqrt{10}}{3}}}}}\:\orange{\bigstar}$