Question

Rationalisation of the denominator of 1÷(√5+√2) gives:

1/√10
√5 + √2
√5 - √2
(√5 - √2)/3​

Answer 1

Question :-

Rationalisation of denominator of $\sf{\dfrac{1}{\sqrt{5} + \sqrt{2}}}$ gives:

$\bull \: \: \: \: \sf{\dfrac{1}{\sqrt{10}}}$

$\bull \: \: \: \sf{ \sqrt{5} + \sqrt{2}}$

$\bull \: \: \: \sf{ \sqrt{5} - \sqrt{2}}$

$\bull \: \: \: \: \sf{\dfrac{ \sqrt{5} - \sqrt{2}}{3}}$

Answer:

Option. d is the correct option.

$\sf{\dfrac{ \sqrt{5} - \sqrt{2}}{3}}$

Step-by-step explanation:

To rationalise

• $\sf{\dfrac{1}{\sqrt{5} + \sqrt{2}}}$

★ Solution

$\sf{\longrightarrow \: \dfrac{1}{\sqrt{5} + \sqrt{2}}}$

Rationalising with the rationalising factor

[√5 - √2], We get.

$\sf{\longrightarrow \: \dfrac{1}{\sqrt{5} + \sqrt{2}} \times \dfrac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}}$

$\sf{\longrightarrow \: \dfrac{ \sqrt{5} - \sqrt{2}}{{(\sqrt{5})}^{2} + ({\sqrt{2})}^{2}}}$

By simplifying, [(√5)² = 5] and [(√2)² = 2], We get.

$\sf{\longrightarrow \: \dfrac{ \sqrt{5} - \sqrt{2}}{5 - 2}}$

$\sf{\longrightarrow \: \dfrac{ \sqrt{5} - \sqrt{2}}{3}}$

_________________________

Hence,

$\sf{\dfrac{1}{\sqrt{5} + \sqrt{2}} = \dfrac{ \sqrt{5} - \sqrt{2}}{3}}$

Answer 2

$\\ \tt \longmapsto \: \frac{1}{ \sqrt{5} + \sqrt{2} } \\ \\ \tt \longmapsto \: \frac{1( \sqrt{5} - \sqrt{2} )}{( \sqrt{5} + \sqrt{2})( \sqrt{5} - \sqrt{2}) } \\ \\ \tt \longmapsto \: \frac{ \sqrt{5} - \sqrt{2} }{( \sqrt{5} {)}^{2 } - ( \sqrt{2} ) {}^{2} } \\ \\ \tt \longmapsto \: \frac{ \sqrt{5} - \sqrt{2} }{5 - 2} \\ \\ \tt \longmapsto \: \frac{ \sqrt{5} - \sqrt{2} }{3} \\ \\ \therefore \sf \: rationalised \: denominator \: is \: \frac{ \sqrt{5} - \sqrt{2} }{3}$