Question

rationalize the dinominator...​

Answer 1

$\frac{1}{ \sqrt{4} + \sqrt{5} } + \frac{1}{ \sqrt{5} + \sqrt{6} } + \frac{1}{ \sqrt{6} + \sqrt{7} } + \frac{1}{ \sqrt{7} + \sqrt{8} } + \frac{1}{ \sqrt{8} + \sqrt{9} }$

$\implies\frac{1}{ \sqrt{5} + \sqrt{4} } + \frac{1}{ \sqrt{6} + \sqrt{5} } + \frac{1}{ \sqrt{7} + \sqrt{6} } + \frac{1}{ \sqrt{8} + \sqrt{7} } + \frac{1}{ \sqrt{9} + \sqrt{8} }$

$\implies\frac{ \sqrt{5} - \sqrt{4} }{ (\sqrt{5} + \sqrt{4})( \sqrt{5} - \sqrt{4)} } + \frac{ \sqrt{6} - \sqrt{5} }{ (\sqrt{6} + \sqrt{5}) ( \sqrt{6} - \sqrt{5} ) } + \frac{ \sqrt{7} - \sqrt{6} }{ (\sqrt{7} + \sqrt{6}) ( \sqrt{7} - \sqrt{6}) } + \frac{ \sqrt{8} - \sqrt{7} }{( \sqrt{8} + \sqrt{7})( \sqrt{8} - \sqrt{7}) } + \frac{ \sqrt{9} - \sqrt{8} }{ (\sqrt{9} + \sqrt{8})( \sqrt{9} - \sqrt{8}) }$

$\implies\frac{ \sqrt{5} - \sqrt{4} }{ (\sqrt{5})^{2} - (\sqrt{4})^{2} } + \frac{ \sqrt{6} - \sqrt{5} }{ {( \sqrt{6} )}^{2} - {( \sqrt{5} )}^{2} } + \frac{ \sqrt{7} - \sqrt{6} }{ {( \sqrt{7} )}^{2} - {(6)}^{2} } + \frac{ \sqrt{8} - \sqrt{7} }{ {( \sqrt{8} )}^{2} - {( \sqrt{7} )}^{2} } + \frac{ \sqrt{9} - \sqrt{8} }{ {( \sqrt{9} )}^{2} - {( \sqrt{8} )}^{2} }$

$\implies\frac{ \sqrt{5} - \sqrt{4} }{ 5 - 4 } + \frac{ \sqrt{6} - \sqrt{5} }{ 6 - 5 } + \frac{ \sqrt{7} - \sqrt{6} }{ 7 - 6 } + \frac{ \sqrt{8} - \sqrt{7} }{ 8 - 7 } + \frac{ \sqrt{9} - \sqrt{8} }{ 9 - 8 }$

$\implies\frac{ \sqrt{5} - \sqrt{4} }{ 1 } + \frac{ \sqrt{6} - \sqrt{5} }{ 1 } + \frac{ \sqrt{7} - \sqrt{6} }{ 1 } + \frac{ \sqrt{8} - \sqrt{7} }{ 1 } + \frac{ \sqrt{9} - \sqrt{8} }{ 1 }$

$\implies \cancel{\sqrt{5} } - \sqrt{4} + \cancel{ \sqrt{6} }- \cancel{\sqrt{5}} + \cancel { \sqrt{7} }- \cancel{\sqrt{6} } + \cancel{\sqrt{8} }- \cancel{\sqrt{7} } + \sqrt{9} - \cancel{\sqrt{8} }$

$\red{ \therefore \sqrt{9} - \sqrt{4} }$

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