Question

Simplify -

$\\ \sf \dfrac{6}{ 2\sqrt{3} \: - \: 6} \: \: + \: \: \dfrac{ \sqrt{6} }{ \sqrt{3} \: + \: \sqrt{2} } \: \: - \: \: \dfrac{4 \sqrt{3}}{ \sqrt{6} \: - \: \sqrt{2} }$
Note -
1. Irrelevant answers will be deleted on the spot!

2. Give answer with steps.

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Answer 1

★ Concept :-

Here the concept of rationalisation has been used. We see that we are given a equation to do. This equation has many rooted terms in the fractions as denominator. So firstly we can rationalise those factorial terms seperately. Then after that we can include those rationalised terms in the main equation and thus find the final answer.

Let's do it !!

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★ Solution :-

Given,

$\;\bf{\mapsto\;\;\red{\dfrac{6}{2\sqrt{3}\:-\:6}\;+\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;-\;\dfrac{4\sqrt{3}}{\sqrt{6}\:-\:\sqrt{2}}}}$

This can be written as ,

$\;\bf{\rightarrow\;\;\dfrac{6}{2(\sqrt{3}\:-\:3)}\;+\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;-\;\dfrac{4\sqrt{3}}{\sqrt{6}\:-\:\sqrt{2}}}$

$\;\bf{\rightarrow\;\;\dfrac{3}{\sqrt{3}\:-\:3}\;+\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;-\;\dfrac{4\sqrt{3}}{\sqrt{6}\:-\:\sqrt{2}}}$

Now let's rationalise each term seperately.

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~ For rationalisation of different terms ::

• First term ::

$\;\sf{\rightarrow\;\;\dfrac{3}{\sqrt{3}\:-\:3}\;=\;\dfrac{3}{\sqrt{3}\:-\:3}\:\times\:\dfrac{\sqrt{3}\:+\:3}{\sqrt{3}\:+\:3}}$

$\;\sf{\rightarrow\;\;\dfrac{3}{\sqrt{3}\:-\:3}\;=\;\dfrac{3(\sqrt{3}\:+\:3)}{(\sqrt{3}\:-\:3)(\sqrt{3}\:+\:3)}}$

Then we know that, (a + b)(a - b) = a² - b²

Using this identity here, we get

$\;\sf{\rightarrow\;\;\dfrac{3}{\sqrt{3}\:-\:3}\;=\;\dfrac{3(\sqrt{3}\:+\:3)}{(\sqrt{3})^{2}\:-\:(3)^{2}}}$

$\;\sf{\rightarrow\;\;\dfrac{3}{\sqrt{3}\:-\:3}\;=\;\dfrac{3(\sqrt{3}\:+\:3)}{3\:-\:9}}$

$\;\sf{\rightarrow\;\;\dfrac{3}{\sqrt{3}\:-\:3}\;=\;\dfrac{3(\sqrt{3}\:+\:3)}{-6}}$

$\;\sf{\rightarrow\;\;\green{\dfrac{3}{\sqrt{3}\:-\:3}\;=\;-\:\dfrac{3(\sqrt{3}\:+\:3)}{6}}}$

• Second Term ::

$\;\sf{\rightarrow\;\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;=\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\:\times\:\dfrac{\sqrt{3}\:-\:\sqrt{2}}{\sqrt{3}\:-\:\sqrt{2}}}$

$\;\sf{\rightarrow\;\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;=\;\dfrac{\sqrt{6}(\sqrt{3}\:-\:\sqrt{2})}{(\sqrt{3}\:+\:\sqrt{2})(\sqrt{3}\:-\:\sqrt{2})}}$

Again using the same identity, we get

$\;\sf{\rightarrow\;\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;=\;\dfrac{\sqrt{18}\:-\:\sqrt{12}}{(\sqrt{3})^{2}\:-\:(\sqrt{2})^{2}}}$

$\;\sf{\rightarrow\;\;\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;=\;\dfrac{3\sqrt{2}\:-\:2\sqrt{3}}{3\:-\:2}}$

$\;\sf{\rightarrow\;\;\blue{\dfrac{\sqrt{6}}{\sqrt{3}\:+\:\sqrt{2}}\;=\;\dfrac{3\sqrt{2}\:-\:2\sqrt{3}}{1}}}$

• Third Term ::

$\;\sf{\rightarrow\;\;\dfrac{4\sqrt{6}}{\sqrt{6}\:-\:\sqrt{2}}\;=\;\dfrac{4\sqrt{3}}{\sqrt{6}\:-\:\sqrt{2}}\:\times\:\dfrac{\sqrt{6}\:+\:\sqrt{2}}{\sqrt{6}\:+\:\sqrt{2}}}$

$\;\sf{\rightarrow\;\;\dfrac{4\sqrt{6}}{\sqrt{6}\:-\:\sqrt{2}}\;=\;\dfrac{4\sqrt{3}(\sqrt{6}\:+\:\sqrt{2})}{(\sqrt{6}\:-\:\sqrt{2})(\sqrt{6}\:+\:\sqrt{2})}}$

Again using same Identity, we get

$\;\sf{\rightarrow\;\;\dfrac{4\sqrt{6}}{\sqrt{6}\:-\:\sqrt{2}}\;=\;\dfrac{4\sqrt{18}\:+\:4\sqrt{6}}{(\sqrt{6})^{2}\:-\:(\sqrt{2})^{2}}}$

$\;\sf{\rightarrow\;\;\dfrac{4\sqrt{6}}{\sqrt{6}\:-\:\sqrt{2}}\;=\;\dfrac{4(3)\sqrt{2}\:+\:4\sqrt{6}}{6\:-\:2}}$

$\;\sf{\rightarrow\;\;\dfrac{4\sqrt{6}}{\sqrt{6}\:-\:\sqrt{2}}\;=\;\dfrac{12\sqrt{2}\:+\:4\sqrt{6}}{4}}$

$\;\sf{\rightarrow\;\;\dfrac{4\sqrt{6}}{\sqrt{6}\:-\:\sqrt{2}}\;=\;\dfrac{4(3\sqrt{2}\:+\:\sqrt{6})}{4}}$

$\;\sf{\rightarrow\;\;\pink{\dfrac{4\sqrt{6}}{\sqrt{6}\:-\:\sqrt{2}}\;=\;\dfrac{3\sqrt{2}\:+\:\sqrt{6}}{1}}}$

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~ For the final answer ::

Let's apply the equations we got into the main equation.

$\;\bf{\Longrightarrow\;\;-\:\dfrac{3(\sqrt{3}\:+\:3)}{6}\;+\;3\sqrt{2}\:-\:2\sqrt{3}\;-\;3\sqrt{2}\:-\:\sqrt{6}}$

Cancelling the unlike terms, we get

$\;\bf{\Longrightarrow\;\;-\:\dfrac{3(\sqrt{3}\:+\:3)}{3(2)}\;-\;2\sqrt{3}\:-\:\sqrt{6}}$

$\;\bf{\Longrightarrow\;\;-\:\dfrac{\sqrt{3}\:+\:3}{2}\;-\;2\sqrt{3}\:-\:\sqrt{6}}$

Now taking LCM as 2 , we get

$\;\bf{\Longrightarrow\;\;\dfrac{-(\sqrt{3}\:+\:3)\;-\;(2)2\sqrt{3}\:-\:2\sqrt{6}}{2}}$

$\;\bf{\Longrightarrow\;\;\dfrac{-\sqrt{3}\:-\:3\;-\;4\sqrt{3}\:-\:2\sqrt{6}}{2}}$

$\;\bf{\Longrightarrow\;\;\dfrac{-\:3\;-\;5\sqrt{3}\:-\:2\sqrt{6}}{2}}$

Now taking -ve sign as common, we get

$\;\bf{\Longrightarrow\;\;\orange{\dfrac{-(3\;+\;5\sqrt{3}\:+\:2\sqrt{6})}{2}}}$

This is the required answer.

$\;\underline{\boxed{\tt{Required\;\:answer\;=\;\bf{\purple{\dfrac{-(3\;+\;5\sqrt{3}\:+\:2\sqrt{6})}{2}}}}}}$

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★ More to know :-

• (a + b)² = a² + b² + 2ab

• (a - b)² = a² + b² - 2ab

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

• (x + a)(x + b) = x² + (a + b)x + ab

Answer 2

$\mathfrak{\huge{\pink{\underline{\underline{AnSwEr:-}}}}}$

Actually Welcome to the Concept of the Rationalization.

=> The concept of Rationalization revolves around the numbers or terms that are irrational in nature, that have to be rationalized by the multiplication of the Conjugate terms of the Denominators to both the Denominator and Numerator.

=> Conjugate here, refers to the terms that have opposite sign but the terms are same.

=> Since this surd has three terms, here we break them individually and rationalise them such that,atlast we can add to get the final term.

The numerical has been solved in the attached pic.