imc

We find the answer by squaring, then square rooting the expression. $$\begin{array}{rl}& \sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}\\ & \\ =& \sqrt{{\left(\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}\right)}^2}\\ & \\ =& \sqrt{9-6\sqrt{2}+2\sqrt{(9-6\sqrt{2})(9+6\sqrt{2})}+9+6\sqrt{2}}\\ & \\ =& \sqrt{18+2\sqrt{(9-6\sqrt{2})(9+6\sqrt{2})}}\\ & \\ =& \sqrt{18+2\sqrt{9^2-(6\sqrt{2}{)}^2}}\\ & \\ =& \sqrt{18+2\sqrt{81-72}}\\ & \\ =& \sqrt{18+2\sqrt{9}}\\ & \\ =& \sqrt{18+6}\\ & \\ =& \sqrt{24}\\ & \\ =& \boxed{2\sqrt{6}}.\end{array}$$