Question

• Rationalize the denominator in the
following.
$\frac{1}{ \sqrt{7 - \sqrt{6} } }$
$\frac{1}{ \sqrt{5 + \sqrt{2} } }$
​

Answer 1

$\star\:\sf \frac{1}{ \sqrt{7 - \sqrt{6} } }$

$\sf{ \frac{1}{ \sqrt{7} - \sqrt{6} }\times \frac{ \sqrt{7} + \sqrt{6} }{ \sqrt{7} +\sqrt{6} } }$

$\implies\sf{ \frac{ \sqrt{7} + \sqrt{6} }{7 - 6} }$

$\implies \boxed{\sf{ \sqrt{7} +\sqrt{6} }}\orange \bigstar$

_________....

$\star\:\sf\frac{1}{ \sqrt{5 + \sqrt{2} } }$

$\sf{ \frac{1}{5 + \sqrt{2} } }$

$\implies\sf{ \frac{1}{\sqrt {5} + \sqrt{2} } \times \frac{\sqrt {5} - \sqrt{2} }{\sqrt {5} - \sqrt{2} } }$

$\implies\sf{ \frac{\sqrt {5 }- \sqrt{2} }{5 - 2} }$

$\implies \boxed{\sf{ \frac{\sqrt {5 }- \sqrt{2} }{3} }} \orange\bigstar$

Answer 2

$\bigstar$ Correct Question:

Rationalize the denominator in the following:

• $\frac{1}{ \sqrt{7} - \sqrt{6} }$
• $\frac{1}{ \sqrt{5} + \sqrt{2} }$

$\bigstar$ Given:

• $\frac{1}{ \sqrt{7} - \sqrt{6} } \:\:and\:\:</li><li>\frac{1}{ \sqrt{5} + \sqrt{2} }$

$\bigstar$To find:

• To Rationalise the denominators.

$\bigstar$ Solution:

$1. \: \: \: \: \frac{1}{ \sqrt{7} - \sqrt{6} }$

$= \frac{ \sqrt{7} + \sqrt{6} }{( \sqrt{7} - \sqrt{6})( \sqrt{7} + \sqrt{6}) }$

(By using a² - b² = (a + b)(a - b))

$= \frac{ \sqrt{7} + \sqrt{6} }{ ( { \sqrt{7}) }^{2} - { (\sqrt{6} )}^{2} }$

$= \frac{ \sqrt{7} + \sqrt{6} }{7 - 6}$

$= \sqrt{7} + \sqrt{6}$

($\because$ √a × √a = a)

$2. \: \: \: \frac{1}{ \sqrt{5} + \sqrt{2} }$

$= \frac{ \sqrt{5} - \sqrt{2} }{ (\sqrt{5} + \sqrt{2} )( \sqrt{5} - \sqrt{2} ) }$

(By using a² - b² = (a + b)(a - b))

$= \frac{ \sqrt{5} - \sqrt{2} }{ {( \sqrt{5} )}^{2} - { (\sqrt{2} )}^{2} }$

$= \frac{ \sqrt{5} - \sqrt{2} }{5 - 2}$

($\because$ √a × √a = a)

$= \frac{ \sqrt{5} - \sqrt{2} }{3}$

$\bigstar$Answers:

$1. \sqrt{7} + \sqrt{6}$

$2. \: \: \frac{ \sqrt{5} - \sqrt{2} }{3}$