Question

$\frac{ \sqrt{5} }{ \sqrt{3} + 2 } - \frac{3 \sqrt{3} }{ \sqrt{2} + \sqrt{5} } + \frac{2 \sqrt{2} }{ \sqrt{3} + \sqrt{5} } =$
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Answer 1

Answer:

1.

Step-by-step explanation:

Given that:

$\tt \red{\frac{ \sqrt{5} }{ \sqrt{3} + 2} - \frac{3 \sqrt{3} }{ \sqrt{2} + \sqrt{2} } + \frac{2 \sqrt{2} }{ \sqrt{3 } + \sqrt{5} } }$

What to do:

To solve

Solution:

[Hint: we can solve this problem one by one and last we arrange the three term according to the given question and cancel them.]

$First \: term: \frac{ \sqrt{5} }{ \sqrt{3} + 2}$

$\longrightarrow \: \frac{ \sqrt{5} ( \sqrt{3} - \sqrt{2}) }{( \sqrt{3} + \sqrt{2} )( \sqrt{3} - \sqrt{2} )}$

$\longrightarrow \: \frac{ \sqrt{15} - \sqrt{10} }{( \sqrt{3} {)}^{2} - ( \sqrt{2 } {)}^{2} }$

$\longrightarrow \: \frac{ \sqrt{15} - \sqrt{10} }{3 - 2}$

$\longrightarrow \: \red{ \sqrt{15} - \sqrt{10} }$

$Second \: term: \frac{3 \sqrt{3} }{ \sqrt{2} + \sqrt{5} }$

$\longrightarrow \: \frac{3 \sqrt{3} ( \sqrt{5} - \sqrt{2} )}{( \sqrt{5} - \sqrt{2})( \sqrt{5} - \sqrt{2}) }$

$\longrightarrow \: \frac{(3 \sqrt{15} - 3 \sqrt{6} )}{( \sqrt{2}) ^{2} - ( \sqrt{5} {)}^{2} }$

$\longrightarrow \: \frac{3( \sqrt{15} - \sqrt{6}) }{2 - 5}$

$\longrightarrow \: \frac { \cancel 3( \sqrt{15} - \sqrt{6} ) } {\cancel3}$

$\longrightarrow \: \red{ \sqrt{15} + \sqrt{6} }$

$Third \: term: \: \frac{2 \sqrt{2} }{ \sqrt{3} + \sqrt{5} }$

$\longrightarrow \: \frac{2 \sqrt{2} ( \sqrt{5} - \sqrt{3}) }{( \sqrt{5} - \sqrt{3} )( \sqrt{5} + \sqrt{3} }$

$\longrightarrow \: \frac{2 \sqrt{10} - 2 \sqrt{16} }{( \sqrt{3} {)}^{2} - ( \sqrt{5} {)}^{2} }$

$\longrightarrow \: \frac{2( \sqrt{10} - \sqrt{6}) }{3 - 5}$

$\longrightarrow \: \frac{ \cancel2( \sqrt{10} - \sqrt{6} ) }{ \cancel2}$

$\longrightarrow \: \red {\sqrt{10} - \sqrt{6} }$

Now

$Hence,\tt \red{\frac{ \sqrt{5} }{ \sqrt{3} + 2} - \frac{3 \sqrt{3} }{ \sqrt{2} + \sqrt{2} } + \frac{2 \sqrt{2} }{ \sqrt{3 } + \sqrt{5} } }$

$\longrightarrow \: \cancel \red{ \cancel{ \sqrt{15} }- \cancel{\sqrt {10}} - \cancel{\sqrt{15} } + \cancel{\sqrt{6} } + \cancel{\sqrt{10}} - \cancel{\sqrt{6}}}$

$\longrightarrow \red{\: 0 \: Ans.}$