\(\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+...+\dfrac{1}{\sqrt{2008}+\sqrt{2009}}=\dfrac{\sqrt{3}-\sqrt{2}}{3-2}+\dfrac{\sqrt{4}-\sqrt{3}}{4-3}+...+\dfrac{\sqrt{2009}-\sqrt{2008}}{2009-2008}=\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{2009}-\sqrt{2008}=\sqrt{2009}-\sqrt{2}\)
\(\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+...+\dfrac{1}{\sqrt{2008}+\sqrt{2009}}\)
\(=\dfrac{\sqrt{2}-\sqrt{3}}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}+\dfrac{\sqrt{3}-\sqrt{4}}{\left(\sqrt{3}+\sqrt{4}\right)\left(\sqrt{3}-\sqrt{4}\right)}+...+\dfrac{\sqrt{2008}-\sqrt{2009}}{\left(\sqrt{2008}+\sqrt{2009}\right)\left(\sqrt{2008}-\sqrt{2009}\right)}\)
\(=\dfrac{\sqrt{2}-\sqrt{3}}{2-3}+\dfrac{\sqrt{3}-\sqrt{4}}{3-4}+...+\dfrac{\sqrt{2008}-\sqrt{2009}}{2008-2009}\)
\(=-\sqrt{2}+\sqrt{3}-\sqrt{3}+\sqrt{4}-...-\sqrt{2008}+\sqrt{2009}\)
\(=-\sqrt{2}+\sqrt{2009}\)
