$\begin{array}{l} K = \left( {2 - \sqrt 3 } \right)\sqrt {2 + \sqrt 3 } + \left( {2 + \sqrt 3 } \right)\sqrt {2 - \sqrt 3 } \\ \to \sqrt 2 .K = \left( {2 - \sqrt 3 } \right)\sqrt {4 + 2\sqrt 3 } + \left( {2 + \sqrt 3 } \right)\sqrt {4 - 2\sqrt 3 } \\ \to \sqrt 2 K = \left( {2 - \sqrt 3 } \right)\left( {\sqrt 3 + 1} \right) + \left( {2 + \sqrt 3 } \right)\left( {\sqrt 3 - 1} \right)\\ \to \sqrt 2 K = 2\sqrt 3 + 2 - 3 - \sqrt 3 + 2\sqrt 3 - 2 + 3 - \sqrt 3 \\ \to \sqrt 2 K = 2\sqrt 3 \to K = \sqrt 6 \end{array}$
