\(\begin{array}{l}
\sqrt {17 + 12\sqrt 2 } + \sqrt {17 - 12\sqrt 2 } = \sqrt {\left( {9 + 2.3.2\sqrt 2 + {{\left( {2\sqrt 2 } \right)}^2}} \right)} + \sqrt {\left( {9 - 2.3.2\sqrt 2 + {{\left( {2\sqrt 2 } \right)}^2}} \right)} \\
= \sqrt {{{\left( {3 + 2\sqrt 2 } \right)}^2}} + \sqrt {{{\left( {3 - 2\sqrt 2 } \right)}^2}} = 3 + 2\sqrt 2 + 3 - 2\sqrt 2 = 6
\end{array}\)
