Question

Rationalise the denominator of
$\frac{1}{ \sqrt{6} + \sqrt{5} - \sqrt{11} }$


Or
Prove that :
$\frac{1}{2 + \sqrt{3} } + \frac{2}{ \sqrt{5} - \sqrt{3} } + \frac{1}{2 - \sqrt{5} } = 0$
Plzz solve these two questions .
I need help .

Answer 1

HEYA....

SOLUTION IS IN THE ATTACHMENT..


Rationalising the denominator : If the denominator of the expression contains a term with a square root or a number under the radical sign then the process of converting it to an equivalent expression whose denominator is a rational number , is called rationalising the denominator.


•Conjugate of (√a + b) is (√a-b) & Conjugate of (√a - √ b) is (√a + √b).

For Rationalising the denominator , we will multiply the numerator and denominator by conjugate of denominator to remove the radical sign form the denominator.

HOPE THIS WILL HELP YOU...

TYSM........@GOZMIT

Answer 2

Heya here ,,.

Solution ⬇⬇⬇⬇⬇

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Solution of first question
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$question \: \: is \: \frac{1}{ \sqrt{6} + \sqrt{5} - \sqrt{11} } \\ \\ sol. \\ \\ \frac{1}{ \sqrt{6} + \sqrt{5} - \sqrt{11} } \\ \\ = > \frac{1}{( \sqrt{6} + \sqrt{5} ) - \sqrt{11} } \times \frac{( \sqrt{6} + \sqrt{5} ) + \sqrt{11} }{( \sqrt{6} + \sqrt{5} ) + \sqrt{11} } \\ \\ = > \frac{ \sqrt{6} + \sqrt{5} + \sqrt{11} }{( \sqrt{6} + \sqrt{5}) ^{2} - ( { \sqrt{11}) }^{2} } \\ \\ = > \frac{ \sqrt{6} + \sqrt{5} + \sqrt{11} }{6 + 5 + 2 \times \sqrt{6} \times \sqrt{5} - 11} \\ \\ = > \frac{ \sqrt{6} + \sqrt{5} + \sqrt{11} }{2 \sqrt{30 } } \\ \\ = > \frac{ (\sqrt{6} + \sqrt{5} + \sqrt{11}) }{2 \sqrt{30} } \times \frac{ \sqrt{30} }{ \sqrt{30} } \\ \\ = > \frac{( \sqrt{6} + \sqrt{5} + \sqrt{11} ) \sqrt{30} }{60} \\ \\ = > \frac{( \sqrt{6} + \sqrt{5} + \sqrt{11} ) \sqrt{6} \times \sqrt{5} }{60} \\ \\ = > \frac{6 \times \sqrt{5} + 5 \times \sqrt{6} \times \sqrt{330} }{60} \\ answer$
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Solution of second question
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$given \: \: \frac{1}{2 + \sqrt{3} } + \frac{2}{ \sqrt{5} - \sqrt{3} } + \frac{1}{2 - \sqrt{5} } = 0 \\ \\ solution. \\ \\ \frac{1}{2 + \sqrt{3} } = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ = > \frac{2 - \sqrt{3} }{1} = 2 - \sqrt{3} \\ \\ similarly \: \\ \\ \frac{2}{ \sqrt{5} - \sqrt{3} } = \sqrt{5} + \sqrt{3} \\ \\ and \: \: \frac{1}{2 - \sqrt{5} } = - (2 + \sqrt{5} ) \\ \\ given \: \: equation \: \\ = > (2 - \sqrt{3} ) + ( \sqrt{5} + \frac{4}{ \sqrt{3} } ) - (2 + \sqrt{5} ) \\ \\ = > 0 \\ \\ hence \: \: proved$
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Hope it's helps you.
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