2. $\sqrt[4018]{5+2\sqrt{6}}\times \sqrt[2009]{\sqrt{3}-\sqrt{2}}$
=$\sqrt[4018]{(\sqrt{3}+\sqrt{2})^2}\times\sqrt[2009]{\sqrt{3}-\sqrt{2}}$
=$\sqrt[2009]{(\sqrt{3}+\sqrt{2})}\times\sqrt[2009]{\sqrt{3}-\sqrt{2}}$
=$\sqrt[2009]{(\sqrt{3}+\sqrt{2})\times\ (\sqrt{3}-\sqrt{2})}$
= 1
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