\[a^{\sqrt{2}} \cdot a^{\sqrt{8}}\ :a^{\sqrt{18}} =\]
\[= a^{\sqrt{2}} \cdot a^{2\sqrt{2}}\ :a^{3\sqrt{2}} =\]
\[= a^{\sqrt{2} + 2\sqrt{2} - 3\sqrt{2}} = a^{0} = 1.\]