$\begin{array}{l}
\frac{{2 + \sqrt 2 }}{{1 + \sqrt 2 }} = \frac{{\sqrt 2 \left( {\sqrt 2 + 1} \right)}}{{\sqrt 2 + 1}} = \sqrt 2 \\
Cach\,\,2:\\
\frac{{2 + \sqrt 2 }}{{1 + \sqrt 2 }} = \frac{{\left( {2 + \sqrt 2 } \right)\left( {\sqrt 2 - 1} \right)}}{{\left( {1 + \sqrt 2 } \right)\left( {\sqrt 2 - 1} \right)}} = \frac{{2\sqrt 2 - 2 + 2 - \sqrt 2 }}{{2 - 1}} = \sqrt 2 .
\end{array}$
