\[\begin{array}{l}
A = \sqrt {3 - \sqrt 5 } + \sqrt {3 + \sqrt 5 } \\
\Rightarrow \sqrt 2 A = \sqrt {2\left( {3 - \sqrt 5 } \right)} + \sqrt {2\left( {3 + \sqrt 5 } \right)} \\
\Leftrightarrow \sqrt 2 A = \sqrt {6 - 2\sqrt 5 } + \sqrt {6 + 2\sqrt 5 } \\
\Leftrightarrow \sqrt 2 A = \sqrt {{{\left( {\sqrt 5 - 1} \right)}^2}} + \sqrt {{{\left( {\sqrt 5 + 1} \right)}^2}} \\
\Leftrightarrow \sqrt 2 A = \left| {\sqrt 5 - 1} \right| + \left| {\sqrt 5 + 1} \right|\\
\Leftrightarrow \sqrt 2 A = \sqrt 5 - 1 + \sqrt 5 + 1\,\,\,\left( {do\,\,\sqrt 5 - 1 > 0} \right)\\
\Leftrightarrow \sqrt 2 A = 2\sqrt 5 \\
\Leftrightarrow A = \frac{{2\sqrt 5 }}{{\sqrt 2 }} = \sqrt {10} .
\end{array}\]
