Question

Simplify
$\frac{2 + \sqrt{3} } {2 - \sqrt{3} } - \frac{2 - \sqrt{3} }{2 + \sqrt{3} }$
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Answer 1

$\sf given : - \frac{2 + \sqrt{3} }{2 - \sqrt{3} } - \frac{2 - \sqrt{3} }{2 + \sqrt{3} } \\ \\ \small \sf = \frac{(2 + \sqrt{3} )(2 + \sqrt{3}) }{(2 - \sqrt{3})(2 + \sqrt{3} ) } - \frac{(2 - \sqrt{3})(2 - \sqrt{3} )}{(2 - \sqrt{3})(2 + \sqrt{3}) } \\ \\ \sf = \frac{{(2 + \sqrt{3} )}^{2} }{( {2)}^{2} - ( { \sqrt{3} })^{2} } - \frac{ {(2 - \sqrt{3} })^{2} }{( {2)}^{2} - ( { \sqrt{3} )}^{2} } \\ \\ \sf = \frac{( {2)}^{2} + 2(2)( \sqrt{3}) + {( \sqrt{3} })^{2} }{4 - 3} - \\ \sf \frac{( {2)}^{2} - 2(2)( \sqrt{3} ) + ( { \sqrt{3} )}^{2} }{4 - 3} \\ \\ \sf = \frac{4 + 4 \sqrt{3} + 3}{1} - \frac{4 - 4 \sqrt{3} + 3}{1} \\ \\ \sf = \cancel4 + 4 \sqrt{3} + \cancel 3 \cancel{- 4} + 4 \sqrt{3} \cancel {- 3} \\ \\ \sf = \boxed{8 \sqrt{3}}$

identities used :-

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• (a + b)(a - b) = a² - b²

Answer 2

$\dfrac{2 \: + \: \sqrt{3} }{2 \: - \: \sqrt{3} } \: - \: \dfrac{2 \: - \: \sqrt{3} }{2 \: + \: \sqrt{3} }$

____________ [ GIVEN ]

• We have to simply it.. means we have to find the value of x.

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=> $\dfrac{2 \: + \: \sqrt{3} }{2 \: - \: \sqrt{3} } \: - \: \dfrac{2 \: - \: \sqrt{3} }{2 \: + \: \sqrt{3} }$

=> $\frac{2 \: + \: \sqrt{3} }{2 \: - \: \sqrt{3} } \: \times \: \frac{2 \: + \: \sqrt{3} }{2 \: + \: \sqrt{3} } - \: \frac{2 \: - \: \sqrt{3} }{2 \: + \: \sqrt{3} } \: \times \: \frac{2 \: - \: \sqrt{3} }{2 \: - \: \sqrt{3} }$

Now..

(a + b) (a + b) = a² + b² + 2ab

(a - b) (a + b) = a² - b²

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

=> $\dfrac{(2 )^{2} \: + \: (\sqrt{3})^{2} \: + \: 2(2)( \sqrt{3}) }{ 4\: - \: 3 } \: - \: \dfrac{(2 )^{2} \: + \: (\sqrt{3} )^{2} \: - 2(2)( \sqrt{3} ) }{ 4\: - \: 3 }$

=> $\dfrac{(2 )^{2} \: + \: (\sqrt{3})^{2} \: + \: 2(2)( \sqrt{3}) }{ 1 } \: - \: \dfrac{(2 )^{2} \: + \: (\sqrt{3} )^{2} \: - 2(2)( \sqrt{3} ) }{ 1 }$

=> 4 + 3 + 4√3 - (4 + 3 - 4√3)

=> 4 + 3 + 4√3 - 4 - 3 + 4√3

=> 4√3 + 4√3

=> 8√3

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8√3 is the answer.

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