Đáp án:
$\begin{array}{l}
\dfrac{{3 + \sqrt 5 }}{{2\sqrt 2 + \sqrt {3 + \sqrt 5 } }} + \dfrac{{3 - \sqrt 5 }}{{2\sqrt 2 - \sqrt {3 - \sqrt 5 } }}\\
= \sqrt 2 .\left( {\dfrac{{3 + \sqrt 5 }}{{4 + \sqrt {6 + 2\sqrt 5 } }} + \dfrac{{3 - \sqrt 5 }}{{4 - \sqrt {6 - 2\sqrt 5 } }}} \right)\\
= \sqrt 2 .\left( {\dfrac{{3 + \sqrt 5 }}{{4 + \sqrt {{{\left( {\sqrt 5 + 1} \right)}^2}} }} + \dfrac{{3 - \sqrt 5 }}{{4 - {{\left( {\sqrt 5 - 1} \right)}^2}}}} \right)\\
= \sqrt 2 .\left( {\dfrac{{3 + \sqrt 5 }}{{4 + \sqrt 5 + 1}} + \dfrac{{3 - \sqrt 5 }}{{4 - \sqrt 5 + 1}}} \right)\\
= \sqrt 2 .\dfrac{{\left( {3 + \sqrt 5 } \right)\left( {\sqrt 5 - 1} \right) + \left( {3 - \sqrt 5 } \right)\left( {\sqrt 5 + 1} \right)}}{{\sqrt 5 \left( {\sqrt 5 - 1} \right)\left( {\sqrt 5 + 1} \right)}}\\
= \sqrt 2 .\dfrac{{3\sqrt 5 - 3 + 5 - \sqrt 5 + 3\sqrt 5 + 3 - 5 - \sqrt 5 }}{{\sqrt 5 .\left( {5 - 1} \right)}}\\
= \sqrt 2 .\dfrac{{4\sqrt 5 }}{{4\sqrt 5 }}\\
= \sqrt 2
\end{array}$
