Question

Rationalise the following :-

$\\ \\ \large \red{ \frac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } - \frac{ 2\sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \frac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } }$

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Answer 1

Basic Concept Used :-

Method of Rationalization :-

The method of Rationalization is used to remove the radicals from denominator and it means multiply and divide by conjugate of denominator.

$\boxed{ \sf \: (x + y)(x - y) = {x}^{2} - {y}^{2}}$

$\large\underline{\sf{Solution-}}$

$\red{\rm :\longmapsto\: \dfrac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } - \dfrac{ 2\sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \dfrac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } }$

Consider,

$\blue{\rm :\longmapsto\:\dfrac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3}}}$

${ \: \rm = \: \: \dfrac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3}} \times \dfrac{ \sqrt{10} - \sqrt{3} }{ \sqrt{10} - \sqrt{3} } }$

$\rm = \: \: \dfrac{7 \sqrt{30} - 21}{10 - 3}$

$\rm = \: \: \dfrac{7 (\sqrt{30} - 3)}{7}$

$\rm = \: \: \sqrt{30} - 3$

$\: \: \: \: \: \: \: \blue{ \boxed{\bf :\implies\:\dfrac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3}} = \sqrt{30} - 3 }}$

Consider,

$\green{ \: \bf :\longmapsto\:\dfrac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5}}}$

$\rm = \: \: \dfrac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5} } \times \dfrac{ \sqrt{6} - \sqrt{5} }{ \sqrt{6} - \sqrt{5} }$

$\rm = \: \: \dfrac{2 \sqrt{30} - 10}{6 - 5}$

$\rm = \: \: 2 \sqrt{30} - 10$

$\: \: \: \: \: \: \: \green{ \boxed{ \: \bf :\implies\:\dfrac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5}} = 2 \sqrt{30} - 10}}$

Consider,

$\purple{ \: \bf :\longmapsto\:\dfrac{3 \sqrt{2} }{ \sqrt{15} + 3\sqrt{2}}}$

$\rm = \: \: \dfrac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } \times \dfrac{ \sqrt{15} - 3 \sqrt{2} }{ \sqrt{15} - 3 \sqrt{2} }$

$\rm = \: \: \dfrac{3 \sqrt{30} - 18 }{15 - 18}$

$\rm = \: \: \dfrac{3( \sqrt{30} - 6)}{ - 3}$

$\rm = \: \: 6 - \sqrt{30}$

$\: \: \: \: \: \: \: \: \purple{ \boxed{\:\bf :\implies\:\dfrac{3 \sqrt{2} }{ \sqrt{15} + 3\sqrt{2}} = 6 - \sqrt{30}}}$

Therefore,

$\red{\rm :\longmapsto\: \dfrac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } - \dfrac{ 2\sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \dfrac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } }$

$\rm = \: \: \sqrt{30} - 3 - (2 \sqrt{30} - 10) - (6 - \sqrt{30})$

$\rm = \: \: \sqrt{30} - 3 - 2 \sqrt{30} + 10 - 6 + \sqrt{30})$

$\rm = \: \: 2 \sqrt{30} - 2 \sqrt{30} + 1$

$\: \rm = \: \: 1$

Hence,

$\red{ \boxed{ \dfrac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } - \dfrac{ 2\sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \dfrac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } = 1}}$

Answer 2

$\: \: \: \: \large\pink꧁ \green{{\underline{\red{ \underline{\purple{\overline{\orange{\overline{ \huge\blue{AnsweR}}}}}}}}}} \pink꧂$