\[\begin{array}{l}
A = \frac{{\sqrt {45 + 27\sqrt 2 } + \sqrt {45 - 27\sqrt 2 } }}{{\sqrt {5 + 3\sqrt 2 } - \sqrt {5 - 3\sqrt 2 } }} + \frac{{\sqrt {3 + \sqrt 2 } + \sqrt {3 - \sqrt 2 } }}{{\sqrt {3 + \sqrt 2 } - \sqrt {3 - \sqrt 2 } }}\\
= \frac{{\sqrt {9\left( {5 + 3\sqrt 2 } \right)} + \sqrt {9\left( {5 - 3\sqrt 2 } \right)} }}{{\sqrt {5 + 3\sqrt 2 } - \sqrt {5 - 3\sqrt 2 } }} + \frac{{\sqrt {3 + \sqrt 2 } + \sqrt {3 - \sqrt 2 } }}{{\sqrt {3 + \sqrt 2 } - \sqrt {3 - \sqrt 2 } }}\\
= \frac{{3\left( {\sqrt {5 + 3\sqrt 2 } + \sqrt {5 - 3\sqrt 2 } } \right)}}{{\sqrt {5 + 3\sqrt 2 } - \sqrt {5 - 3\sqrt 2 } }} + \frac{{{{\left( {\sqrt {3 + \sqrt 2 } + \sqrt {3 - \sqrt 2 } } \right)}^2}}}{{3 + \sqrt 2 - 3 + \sqrt 2 }}\\
= \frac{{3{{\left( {\sqrt {5 + 3\sqrt 2 } + \sqrt {5 - 3\sqrt 2 } } \right)}^2}}}{{5 + 3\sqrt 2 - 5 + 3\sqrt 2 }} + \frac{{3 + \sqrt 2 + 3 - \sqrt 2 + 2\sqrt {\left( {3 + \sqrt 2 } \right)\left( {3 - \sqrt 2 } \right)} }}{{2\sqrt 2 }}\\
= \frac{{3\left( {5 + 3\sqrt 2 + 5 - 3\sqrt 2 + 2\sqrt {\left( {5 + 3\sqrt 2 } \right)\left( {5 - 3\sqrt 2 } \right)} } \right)}}{{6\sqrt 2 }} + \frac{{6 + 2\sqrt {9 - 2} }}{{2\sqrt 2 }}\\
= \frac{{3\left( {10 + 2\sqrt {25 - 18} } \right)}}{{6\sqrt 2 }} + \frac{{3 + \sqrt 7 }}{{\sqrt 2 }}\\
= \frac{{5 + \sqrt 7 }}{{\sqrt 2 }} + \frac{{3 + \sqrt 7 }}{{\sqrt 2 }} = \frac{{5 + \sqrt 7 + 3 + \sqrt 7 }}{{\sqrt 2 }}\\
= \frac{{8 + 2\sqrt 7 }}{{\sqrt 2 }} = 2\sqrt 2 + \sqrt {14} .
\end{array}\]
