$\begin{array}{l} 1)2\sqrt {9 - 4\sqrt 2 } + 2\sqrt {3 + 2\sqrt 2 } \\ = 2\sqrt {8 + 1 - 2.2\sqrt 2 } + 2\sqrt {2 + 1 + 2\sqrt 2 } \\ = 2\sqrt {{{\left( {2\sqrt 2 - 1} \right)}^2}} + 2\sqrt {{{\left( {\sqrt 2 + 1} \right)}^2}} \\ = 2\left( {2\sqrt 2 - 1} \right) + 2\left( {\sqrt 2 + 1} \right)\\ = 4\sqrt 2 - 2 + 2\sqrt 2 + 2 = 6\sqrt 2 \\ 2)\sqrt {11 + 6\sqrt 2 } - 3\sqrt {{{\left( {2\sqrt 2 - 3} \right)}^2}} + \dfrac{{2 + \sqrt 2 }}{{1 + \sqrt 2 }}\\ = \sqrt {9 + 2 + 2.3\sqrt 2 } - 3\left( {3 - 2\sqrt 2 } \right) + \dfrac{{\sqrt 2 \left( {\sqrt 2 + 1} \right)}}{{1 + \sqrt 2 }}\\ = \sqrt {{{\left( {3 + \sqrt 2 } \right)}^2}} - 3\left( {3 - 2\sqrt 2 } \right) + \sqrt 2 \\ = 3 + \sqrt 2 - 9 + 6\sqrt 2 + \sqrt 2 = 8\sqrt 2 - 6\\ 3)\left( {\dfrac{1}{{\sqrt 5 - 2}} - \dfrac{{59}}{{3\sqrt 7 - 2}}} \right)\left( {\sqrt 5 + 3\sqrt 7 } \right)\\ = \left( {\dfrac{{\sqrt 5 + 2}}{{5 - 4}} - \dfrac{{59\left( {3\sqrt 7 + 2} \right)}}{{9.7 - 4}}} \right)\left( {\sqrt 5 + 3\sqrt 7 } \right)\\ = \left( {\sqrt 5 + 2 - 3\sqrt 7 - 2} \right)\left( {\sqrt 5 + 3\sqrt 7 } \right)\\ = \left( {\sqrt 5 - 3\sqrt 7 } \right)\left( {\sqrt 5 + 3\sqrt 7 } \right) = 5 - 9.7 = - 58\\ 4)\dfrac{{2\sqrt {3 + \sqrt {5 - \sqrt {13 + \sqrt {48} } } } }}{{\sqrt 6 + \sqrt 2 }}\\ = \dfrac{{2\sqrt {3 + \sqrt {5 - \sqrt {13 + 4\sqrt 3 } } } }}{{\sqrt 6 + \sqrt 2 }}\\ = \dfrac{{2\sqrt {3 + \sqrt {5 - \sqrt {{{\left( {2\sqrt 3 + 1} \right)}^2}} } } }}{{\sqrt 6 + \sqrt 2 }}\\ = \dfrac{{2\sqrt {3 + \sqrt {5 - 2\sqrt 3 - 1} } }}{{\sqrt 6 + \sqrt 2 }}\\ = \dfrac{{2\sqrt {3 + \sqrt {4 - 2\sqrt 3 } } }}{{\sqrt 6 + \sqrt 2 }} = \dfrac{{2\sqrt {3 + \sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} } }}{{\sqrt 6 + \sqrt 2 }}\\ = \dfrac{{2\sqrt {2 + \sqrt 3 } }}{{\sqrt 6 + \sqrt 2 }} = \dfrac{{\sqrt 2 .\sqrt {4 + 2\sqrt 3 } }}{{\sqrt 2 \left( {1 + \sqrt 3 } \right)}} = \dfrac{{\sqrt {{{\left( {\sqrt 3 + 1} \right)}^2}} }}{{1 + \sqrt 3 }}\\ = \dfrac{{\sqrt 3 + 1}}{{1 + \sqrt 3 }} = 1 \end{array}$
