\( \dfrac{{2 - \sqrt 3 }}{{\sqrt 2 + \sqrt {2 + \sqrt 3 } }} + \dfrac{{2 + \sqrt 3 }}{{\sqrt 2 - \sqrt {2 - \sqrt 3 } }}\\ = \dfrac{{\sqrt 2 \left( {2 - \sqrt 3 } \right)}}{{\sqrt 2 \left( {\sqrt 2 + \sqrt {2 + \sqrt 3 } } \right)}} + \dfrac{{\sqrt 2 \left( {2 + \sqrt 3 } \right)}}{{\sqrt 2 \left( {\sqrt 2 - \sqrt {2 - \sqrt 3 } } \right)}}\\ = \dfrac{{2\sqrt 2 - \sqrt 6 }}{{2 + \sqrt {2\left( {2 + \sqrt 3 } \right)} }} + \dfrac{{2\sqrt 2 + \sqrt 6 }}{{2 - \sqrt {2\left( {2 - \sqrt 3 } \right)} }}\\ = \dfrac{{2\sqrt 2 - \sqrt 6 }}{{2 + \sqrt {4 + 2\sqrt 3 } }} + \dfrac{{2\sqrt 2 + \sqrt 6 }}{{2 - \sqrt {4 - 2\sqrt 3 } }}\\ = \dfrac{{2\sqrt 2 - \sqrt 6 }}{{2 + \sqrt {{{\left( {1 + \sqrt 3 } \right)}^2}} }} + \dfrac{{2\sqrt 2 + \sqrt 6 }}{{2 - \sqrt {{{\left( {1 - \sqrt 3 } \right)}^2}} }}\\ = \dfrac{{2\sqrt 2 - \sqrt 6 }}{{2 + 1 + \sqrt 3 }} + \dfrac{{2\sqrt 2 + \sqrt 6 }}{{2 - \sqrt 3 + 1}}\\ = \dfrac{{2\sqrt 2 - \sqrt 6 }}{{3 + \sqrt 3 }} + \dfrac{{2\sqrt 2 + \sqrt 6 }}{{3 - \sqrt 3 }}\\ = \dfrac{{\left( {2\sqrt 2 - \sqrt 6 } \right)\left( {3 - \sqrt 3 } \right)}}{{\left( {3 + \sqrt 3 } \right)\left( {3 - \sqrt 3 } \right)}} + \dfrac{{\left( {2\sqrt 2 + \sqrt 6 } \right)\left( {3 + \sqrt 3 } \right)}}{{\left( {3 - \sqrt 3 } \right)\left( {3 + \sqrt 3 } \right)}}\\ = \dfrac{{6\sqrt 2 - 2\sqrt 6 - 3\sqrt 6 + 3\sqrt 2 }}{6} + \dfrac{{6\sqrt 2 + 2\sqrt 6 + 3\sqrt 6 + 3\sqrt 2 }}{6}\\ = \dfrac{{6\sqrt 2 - 5\sqrt 6 + 3\sqrt 2 + 6\sqrt 2 + 5\sqrt 6 + 3\sqrt 2 }}{6}\\ = \dfrac{{18\sqrt 2 }}{6} = 3\sqrt 2 \)
