Answer:
[tex]\boxed{\bf\:x + y = 4 \sqrt{2} - 2 \sqrt{3} \: } \\ [/tex]
Identity used:
[tex]\boxed{\sf\:(x + y)(x - y) = {x}^{2} - {y}^{2} \: } \\ [/tex]
Step-by-step explanation:
Given that,
[tex]\sf\: x = \dfrac{2 + \sqrt{3} }{ \sqrt{2} + 1} \\ [/tex]
[tex]\sf\: y = \dfrac{2 - \sqrt{3} }{ \sqrt{2} - 1} \\ [/tex]
Now, Consider
[tex]\sf\: x + y \\ [/tex]
[tex]\sf\: = \: \dfrac{2 + \sqrt{3} }{ \sqrt{2} + 1} + \dfrac{2 - \sqrt{3}}{ \sqrt{2} - 1 } \\ [/tex]
[tex]\sf\: = \: \dfrac{(2 + \sqrt{3})( \sqrt{2} - 1) + (2 - \sqrt{3})( \sqrt{2} + 1) }{ (\sqrt{2} + 1)( \sqrt{2} - 1)} \\ [/tex]
[tex]\sf\: = \: \dfrac{2( \sqrt{2} - 1) + \sqrt{3}( \sqrt{2} - 1) + 2( \sqrt{2} + 1) - \sqrt{3}( \sqrt{2} + 1)}{ (\sqrt{2})^{2} - {(1)}^{2} } \\ [/tex]
[tex]\sf\: = \: \dfrac{2( \sqrt{2} - 1 + \sqrt{2} + 1 ) + \sqrt{3}( \sqrt{2} - 1 - \sqrt{2} - 1)}{ 2 - 1 } \\ [/tex]
[tex]\sf\: = \: \dfrac{2( 2\sqrt{2}) + \sqrt{3}( - 2)}{ 1 } \\ [/tex]
[tex]\sf\: = \: 4 \sqrt{2} - 2 \sqrt{3} \\ [/tex]
Hence,
[tex]\implies\sf\:\boxed{\bf\:x + y = 4 \sqrt{2} - 2 \sqrt{3} \: } \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional information:
[tex]\sf \: {( \sqrt{x} + \sqrt{y})}^{2} = x + y + 2 \sqrt{xy} \\ [/tex]
[tex]\sf \: {( \sqrt{x} - \sqrt{y})}^{2} = x + y - 2 \sqrt{xy} \\ [/tex]
[tex]\sf \: {( \sqrt{x} + \sqrt{y}) }^{2} + {( \sqrt{x} - \sqrt{y})}^{2} = 2(x + y) \\ [/tex]
[tex]\sf \: {( \sqrt{x} + \sqrt{y}) }^{2} - {( \sqrt{x} - \sqrt{y})}^{2} = 4 \sqrt{xy} \\ [/tex]
[tex]\sf \: \sqrt{x} \times \sqrt{y} = \sqrt{xy} \\ [/tex]
[tex]\sf \: \sqrt{x} + \sqrt{y} \: \ne \: \sqrt{x + y} \\ [/tex]
[tex]\sf \: \sqrt{x} - \sqrt{y} \: \ne \: \sqrt{x - y} \\ [/tex]
[tex]\sf \: \sqrt{ {x}^{2} + {y}^{2} } \: \ne \: x + y \\ [/tex]
[tex]\sf \: \sqrt{ {x}^{2} - {y}^{2} } \: \ne \: x - y [/tex]
