Question

Simplify by rationalizing the denominator . Root5-2/root5+2 -root5+2/root5-2

Answer 1

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

$\sf \longmapsto \dfrac{ \sqrt{5} - 2}{ \sqrt{5} + 2} - \dfrac{ \sqrt{5} + 2 }{ \sqrt{5} - 2}$

Rationalising means replacing root from denominator to the numerator to make the term more simplified.

=>In rationalising we multiply with opposite sign of denominate to both numerator and denominator.

$\sf \longmapsto \dfrac{ \sqrt{5} - 2}{ \sqrt{5} + 2 } \times \dfrac{ \sqrt{5} - 2 }{ \sqrt{5} - 2} - \dfrac{ \sqrt{5} + 2}{ \sqrt{5} - 2} \times \dfrac{ \sqrt{5} + 2}{ \sqrt{5} + 2 }$

Identity apply here :

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

$\sf \bold{ {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy}$

$\sf \bold{ {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}$

$\sf \longmapsto \dfrac{ {( \sqrt{5} - 2)}^{2} }{ {( \sqrt{5} )}^{2} - {(2)}^{2} } - \dfrac{ {( \sqrt{5} + 2)}^{2} }{ {( \sqrt{5} )}^{2} - {(2)}^{2} }$

$\sf \longmapsto \dfrac{ {( \sqrt{5}) }^{2} + {(2)}^{2} - 2( \sqrt{5} )(2) }{5 - 4} - \dfrac{ {( \sqrt{5} )}^{2} + {(2)}^{2} + 2( \sqrt{5})(2) }{5 - 4}$

$\sf \longmapsto5 + 4 - 4 \sqrt{5} - (5 + 4 + 4 \sqrt{5} )$

$\sf \longmapsto9 - 4 \sqrt{5} - 9 - 4 \sqrt{5}$

$\sf \longmapsto{\cancel{9}} - 4 \sqrt{5} - {\cancel{9}} -4 \sqrt{5} )$

=>-8√5

Answer 2

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Step-by-step explanation:

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