Question

simplify the following​

Answer 1

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

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Answer 2

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

★ Given

$\rm :\longmapsto\:\dfrac{1}{2 + \sqrt{3} } + \dfrac{2}{ \sqrt{5} - \sqrt{3} } + \dfrac{1}{2 - \sqrt{5} } - - (1)$

Consider,

$\red{\bf :\longmapsto\:\dfrac{1}{2 + \sqrt{3}}}$

★ On rationalizing the denominator, we get

$\rm \: \: = \: \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} }$

$\rm \: \: = \: \dfrac{2 - \sqrt{3} }{ {2}^{2} - {( \sqrt{3} ) \: }^{2} }$

$\: \: \: \: \: \: \red{\bigg \{ \because \:(x + y)(x - y) = {x}^{2} - {y}^{2} \bigg \}}$

$\rm \: \: = \: \dfrac{2 - \sqrt{3} }{4 - 3}$

$\rm \: \: = \: \dfrac{2 - \sqrt{3} }{1}$

$\rm \: \: = \: 2 - \sqrt{3}$

$\red{\bf :\longmapsto\:\dfrac{1}{2 + \sqrt{3}} = 2 - \sqrt{3} - - - (2) }$

Consider,

$\blue{ \bf{ \longmapsto \: \dfrac{2}{ \sqrt{5} - \sqrt{3} } }}$

★ On rationalizing the denominator, we get

$\rm \: \: = \: \dfrac{2}{ \sqrt{5} - \sqrt{3} } \times \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} }$

$\rm \: \: = \: \dfrac{2( \sqrt{5} + \sqrt{3})}{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} }$

$\: \: \: \: \: \: \blue{\bigg \{ \because \:(x + y)(x - y) = {x}^{2} - {y}^{2} \bigg \}}$

$\rm \: \: = \: \dfrac{2( \sqrt{5} + \sqrt{3})}{5 - 3}$

$\rm \: \: = \: \dfrac{2( \sqrt{5} + \sqrt{3})}{2}$

$\rm \: \: = \: \sqrt{5} + \sqrt{3}$

$\blue{ \bf{ \longmapsto \: \dfrac{2}{ \sqrt{5} - \sqrt{3} } } = \sqrt{5} + \sqrt{3} - - (3) }$

Consider,

$\purple{ \bf{ \longmapsto \: \dfrac{1}{2 - \sqrt{5} } }}$

★ On rationalizing the denominator, we get

$\rm \: \: = \: \dfrac{1}{2 - \sqrt{5} } \times \dfrac{2 + \sqrt{5} }{2 + \sqrt{5} }$

$\rm \: \: = \: \dfrac{2 + \sqrt{5} }{ {2}^{2} - {( \sqrt{5} )}^{2} }$

$\: \: \: \: \: \: \purple{\bigg \{ \because \:(x + y)(x - y) = {x}^{2} - {y}^{2} \bigg \}}$

$\rm \: \: = \: \dfrac{2 + \sqrt{5} }{4 - 5}$

$\rm \: \: = \: \dfrac{2 + \sqrt{5} }{ - 1}$

$\rm \: \: = \: - 2 - \sqrt{5}$

$\purple{ \bf{ \longmapsto \: \dfrac{1}{2 - \sqrt{5} } } = - 2 - \sqrt{5} - - - (4)}$

★ Now, Substituting all the values from equation (2), (3) and (4) in equation (1), we get

$\green{\bf :\longmapsto\:\dfrac{1}{2 + \sqrt{3} } + \dfrac{2}{ \sqrt{5} - \sqrt{3} } + \dfrac{1}{2 - \sqrt{5} } }$

$\rm \: \: = \: 2 - \sqrt{3} + \sqrt{5} + \sqrt{3} - 2 - \sqrt{5}$

$\rm \: \: = \: 0$

Hence,

$\green{\bf :\longmapsto\:\dfrac{1}{2 + \sqrt{3} } + \dfrac{2}{ \sqrt{5} - \sqrt{3} } + \dfrac{1}{2 - \sqrt{5} } = 0 }$

More Identities to know:

★ (a + b)² = a² + 2ab + b²

★ (a - b)² = a² - 2ab + b²

★ a² - b² = (a + b)(a - b)

★ (a + b)² = (a - b)² + 4ab

★ (a - b)² = (a + b)² - 4ab

★ (a + b)² + (a - b)² = 2(a² + b²)

★ (a + b)³ = a³ + b³ + 3ab(a + b)

★ (a - b)³ = a³ - b³ - 3ab(a - b)