[m] (1-\sqrt{2})^2=1-2\sqrt{2}+(\sqrt{2})^2)=1-2\sqrt{2}+2=3-2\sqrt{2}[/m],
то
[m]\sqrt[3]{1+\sqrt{2}}\cdot \sqrt[6]{3-2\sqrt{2}}=\sqrt[3]{1+\sqrt{2}}\cdot \sqrt[6]{(1-\sqrt{2})^2}=\sqrt[3]{1+\sqrt{2}}\cdot \sqrt[3]{1-\sqrt{2}}=[/m]
[m]=\sqrt[3]{(1+\sqrt{2})\cdot (1-\sqrt{2}}=\sqrt[3]{(1)^2-(\sqrt{2})^2}=\sqrt[3]{-1}=-1[/m]