Question

pls ans this ques.....
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Answer 1

Answer:

$\large\boxed{\sf{\dfrac{5 \sqrt{6} + 6 \sqrt{5} + \sqrt{330} }{60} }}$

Step-by-step explanation:

Given a fraction such that,

$\dfrac{1}{ \sqrt{5} + \sqrt{6} - \sqrt{11} }$

Let's rationalize its denominator.

$= \frac{1}{( \sqrt{5} + \sqrt{6}) - \sqrt{11} } \times \frac{( \sqrt{5} + \sqrt{6} ) + \sqrt{11} }{( \sqrt{5} + \sqrt{6}) + \sqrt{11} } \\ \\ = \frac{ \sqrt{5} + \sqrt{6} + \sqrt{11} }{(( \sqrt{5} + \sqrt{6}) + (\sqrt{11} ))(( \sqrt{5} + \sqrt{6}) - ( \sqrt{11} ))} \\ \\ = \frac{ \sqrt{5} + \sqrt{6} + \sqrt{11} }{ {( \sqrt{5} + \sqrt{6}) }^{2} - {( \sqrt{11} )}^{2} } \\ \\ = \frac{ \sqrt{5} + \sqrt{6} + \sqrt{11} }{ {( \sqrt{5}) }^{2} + {( \sqrt{6} )}^{2} + 2( \sqrt{5})( \sqrt{6} ) - {( \sqrt{11} )}^{2} } \\ \\ = \frac{ \sqrt{5} + \sqrt{6} + \sqrt{11} }{5 + 6 + 2 \sqrt{30} - 11} \\ \\ = \frac{ \sqrt{5} + \sqrt{6} + \sqrt{11} }{11 + 2 \sqrt{30} - 11} \\ \\ = \frac{ \sqrt{5} + \sqrt{6} + \sqrt{11} }{2 \sqrt{30} } \\ \\ = \frac{ \sqrt{5} + \sqrt{6} + \sqrt{11} }{2 \sqrt{30} } \times \frac{ \sqrt{30} }{ \sqrt{30} } \\ \\ = \frac{ \sqrt{30} ( \sqrt{5} + \sqrt{6} + \sqrt{11} )}{2 { (\sqrt{30} )}^{2} } \\ \\ = \frac{ \sqrt{150} + \sqrt{180} + \sqrt{330} }{2(30)} \\ \\ = \frac{ \sqrt{25 \times 6} + \sqrt{36 \times 5} + \sqrt{330} }{60 } \\ \\ = \frac{5 \sqrt{6} + 6 \sqrt{5} + \sqrt{330} }{60}$