Question

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Answer 1

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Answer 2

Required Answer:-

Given to simplify:

$\rm \mapsto \dfrac{1}{ \sqrt{7} + \sqrt{6} + \sqrt{13} }$

Solution:

$\rm \dfrac{1}{ \sqrt{7} + \sqrt{6} + \sqrt{13} }$

$\rm = \dfrac{1}{ \sqrt{7} + (\sqrt{6} + \sqrt{13}) }$

$\rm = \dfrac{1 \times ( \sqrt{7} - ( \sqrt{6} + \sqrt{13} ) }{ (\sqrt{7} + (\sqrt{6} + \sqrt{13}))( \sqrt{7} - ( \sqrt{6} + \sqrt{13} )) }$

$\rm = \dfrac{\sqrt{7} - \sqrt{6} - \sqrt{13} }{ (\sqrt{7})^{2} - (\sqrt{6} + \sqrt{13})^{2}}$

$\rm = \dfrac{\sqrt{7} - \sqrt{6} - \sqrt{13} }{ 7 - (6 + 13 + 2 \sqrt{78}) }$

$\rm = \dfrac{\sqrt{7} - \sqrt{6} - \sqrt{13} }{ 7 - ( 19+ 2 \sqrt{78}) }$

$\rm = \dfrac{\sqrt{7} - \sqrt{6} - \sqrt{13} }{ 7 -19 - 2 \sqrt{78} }$

$\rm = \dfrac{\sqrt{7} - \sqrt{6} - \sqrt{13} }{ - 12- 2 \sqrt{78} }$

$\rm = \dfrac{\sqrt{7} - \sqrt{6} - \sqrt{13} }{ -( 12 + 2 \sqrt{78}) }$

$\rm = \dfrac{ \sqrt{6} + \sqrt{13} - \sqrt{7} }{12 + 2 \sqrt{78}}$

$\rm = \dfrac{ (\sqrt{6} + \sqrt{13} - \sqrt{7} )(12 - 2 \sqrt{78}) }{(12 + 2 \sqrt{78})(12 - 2 \sqrt{78} )}$

$\rm = \dfrac{ (\sqrt{6} + \sqrt{13} - \sqrt{7} )(12 - 2 \sqrt{78}) }{{(12)}^{2} - ( 2 \sqrt{78})^{2}}$

$\rm = \dfrac{ (\sqrt{6} + \sqrt{13} - \sqrt{7} )(12 - 2 \sqrt{78}) }{144 -312}$

$\rm = \dfrac{ (\sqrt{6} + \sqrt{13} - \sqrt{7} )(12 - 2 \sqrt{78}) }{ - 168}$

$\rm = \dfrac{\sqrt{6} (12 - 2 \sqrt{78}) + \sqrt{13}(12 - 2 \sqrt{78} ) - \sqrt{7}(12 - 2 \sqrt{78}) }{ - 168}$

$\rm = \dfrac{12 \sqrt{6} - 2 \sqrt{78 \times 6} + 12\sqrt{13}- 2 \sqrt{78 \times 13} - 12\sqrt{7} + 2 \sqrt{78 \times 7} }{ - 168}$

$\rm = \dfrac{12 \sqrt{6} - 2 \sqrt{13\times {6}^{2} } + 12\sqrt{13}- 2 \sqrt{6 \times {13}^{2} } - 12\sqrt{7} + 2 \sqrt{78 \times 7} }{ - 168}$

$\rm = \dfrac{12 \sqrt{6} - 12 \sqrt{13 } + 12\sqrt{13}- 26 \sqrt{6 } - 12\sqrt{7} + 2 \sqrt{78 \times 7} }{ - 168}$

$\rm = \dfrac{12 \sqrt{6}- 26 \sqrt{6 } - 12\sqrt{7} + 2 \sqrt{78 \times 7} }{ - 168}$

$\rm = \dfrac{- 14 \sqrt{6 } - 12\sqrt{7} + 2 \sqrt{78 \times 7} }{ - 168}$

$\rm = \dfrac{2(- 7 \sqrt{6 } - 6\sqrt{7} +\sqrt{78 \times 7} ) }{ - 168}$

$\rm = \dfrac{(- 7 \sqrt{6 } - 6\sqrt{7} +\sqrt{78 \times 7} ) }{ -84}$

$\rm = \dfrac{\sqrt{546}- 6 \sqrt{7} - 7\sqrt{6} }{ -84}$

Hence, this is the simplified form.

Answer:

$\rm \mapsto \dfrac{\sqrt{546}- 6 \sqrt{7} - 7\sqrt{6} }{ -84}$