$(3+\sqrt{2})^{5}=3^{5}+5.3^{4}.\sqrt{2}+10.3^{3}.(\sqrt{2})^{2}+10.3^{2}.(\sqrt{2})^{3}+5.3.(\sqrt{2})^{4}+(\sqrt{2})^{5}\\(3-\sqrt{2})^{5}= 3^{5}-5.3^{4}.\sqrt{2}+10.3^{3}.(\sqrt{2})^{2}-10.3^{2}.(\sqrt{2})^{3}+5.3.(\sqrt{2})^{4}-(\sqrt{2})^{5}\\\Rightarrow (3+\sqrt{2})^{5}-(3-\sqrt{2})^{5}=810\sqrt{2}+360\sqrt{2}+8\sqrt{2}=1178\sqrt{2}$
