Question

ratinalise the denominator √2/√2+√3-5 ​

Answer 1

Given to Rationalize the denominator:-

$\pmb \: \cfrac{ \sqrt{2} }{ \sqrt{2} + \sqrt{3} - 5}$

SOLUTION:-

Rationalization means We have to remove the surds /radicals in denominator by the process of Rationalization. In order to remove the surds we have to multiply and divide with its conjugate of denominator i.e  called as Rationalizing factor .

Conjugate of

$\pmb \ \sqrt{2} + \sqrt{3} - 5 = \sqrt{2} + \sqrt{3} + 5$

So, multiply and divide with this conjugate.

$\pmb \: \dfrac{ \sqrt{2} }{ \sqrt{2} + \sqrt{3} - 5 } \times \dfrac{ \sqrt{2}+\sqrt{3}+5 }{ \sqrt{2} + \sqrt{3} + 5 }$

$\pmb \: \dfrac{( \sqrt{2})( \sqrt{2}+\sqrt{3} +5 ) }{( \sqrt{2} + \sqrt{3} - 5)( \sqrt{2} + \sqrt{3} + 5) }$

Denominator is in form of (a+b)(a-b) = a²-b² Simplifying this

$\pmb \: \dfrac{2+\sqrt{6} +5\sqrt{2} }{( \sqrt{2} + \sqrt{3}) {}^{2} - (5) {}^{2} }$

$\pmb \: \cfrac{2+\sqrt{6} +5\sqrt{2} }{( \sqrt{2}) {}^{2} + ( \sqrt{3} ) {}^{2} + 2( \sqrt{2})( \sqrt{3} ) - 25 }$

$\pmb \: \cfrac{2+\sqrt{6}+5\sqrt{2} }{ 2 + 3 + 2 \sqrt{6} - 25 }$

$\pmb \: \cfrac{2+\sqrt{6}+5\sqrt{2} }{ 5 + 2 \sqrt{6} - 25 }$

$\cfrac{2+\sqrt{6} +5\sqrt{2} }{2 \sqrt{6} - 20 }$

Still the radicals are not removed So, again we have to do the Rationalization.

Rationalizing factor of

$2 \sqrt{6} - 20 \: is \: 2 \sqrt{6} + 20$

So, multiply and divide with this

$\pmb \: \cfrac{2+\sqrt{6}+5\sqrt{2} }{2 \sqrt{6} - 20 } \times \cfrac{2 \sqrt{6} + 20 }{2 \sqrt{6} + 20}$

Now ,in denominator (a-b)(a+b) was formed that is a²-b^2

$\pmb \: \cfrac{(2+\sqrt{6}+5\sqrt{2} )(2\sqrt{6} +20)}{(2 \sqrt{6} - 20)(2\sqrt{6}+20) }$

$\cfrac{4\sqrt{6} +40+12+20\sqrt{6} +10\sqrt{12} +100\sqrt{2} }{(2\sqrt{6})^2-(20)^2 }$

$\cfrac{4\sqrt{6} +52+20\sqrt{6} +10\sqrt{12}+100\sqrt{2} }{(24-400) }$

$\cfrac{4\sqrt{6} +52+20\sqrt{6} +10\sqrt{12}+100\sqrt{2} }{(-376) }$

$\cfrac{2(2\sqrt{6} +26+10\sqrt{6} +5\sqrt{12}+ 50\sqrt{2} )}{(-376) }$

$\cfrac{(2\sqrt{6} +26+10\sqrt{6} +5\sqrt{12}+ 50\sqrt{2} )}{(-188) }$

$\cfrac{(-2\sqrt{6} -26+10\sqrt{6} -5\sqrt{12}- 50\sqrt{2} )}{(188) }$

Since denominator rationalized!