Ta có : \(\frac{2008}{\sqrt{2009}}+\frac{2009}{\sqrt{2008}}\) = \(\frac{2009-1}{\sqrt{2009}}+\frac{2008+1}{\sqrt{2008}}\)
= \(\frac{2009}{\sqrt{2009}}-\frac{1}{\sqrt{2009}}+\frac{2008}{\sqrt{2008}}+\frac{1}{\sqrt{2008}}\)
= \(\frac{\left(\sqrt{2009}\right)^2}{\sqrt{2009}}-\frac{1}{\sqrt{2009}}+\frac{\left(\sqrt{2008}\right)^2}{\sqrt{2008}}+\frac{1}{\sqrt{2008}}\)
= \(\sqrt{2009}-\frac{1}{\sqrt{2009}}+\sqrt{2008}+\frac{1}{\sqrt{2008}}\)
Mà \(\frac{1}{\sqrt{2008}}>\frac{1}{\sqrt{2009}}\)
=> \(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}>0\)
=> \(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}+\sqrt{2008}+\sqrt{2009}>\sqrt{2008}+\sqrt{2009}\)
Vậy \(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}+\sqrt{2008}+\sqrt{2009}>\sqrt{2008}+\sqrt{2009}\) .
