$\begin{align}  & \text{Let }\sqrt{17+12\sqrt{2}}=\sqrt{a}+\sqrt{b} \\ & \text{Square both sides} \\ & a+b+2\sqrt{ab}=17+12\sqrt{2} \\ & \text{Comparing identities} \\ & a+b=17----(i) \\ & b=17-a \\ & 2\sqrt{ab}=12\sqrt{2}---(ii) \\ & ab=72 \\ & \text{Substitute }b=17-a\text{ into }(ii) \\ & (17-a)a=72 \\ & {{a}^{2}}-17a+72=0 \\ & (a-8)(a-9)=0 \\ & a=8\text{ or }a=9 \\ & \text{when }a=8,\text{ }b=17-8=9 \\ & \text{when }a=9,\text{ }b=17-9=8 \\ & \sqrt{17+12\sqrt{2}}=\sqrt{9}+\sqrt{8}\text{ twice }=3+2\sqrt{2}\text{ twice} \\\end{align}$