đkxđ: m≠0, n ≠ 0; mn > 0; m ≠ \(\sqrt{mn}\)
\(\dfrac{m+n}{\sqrt{m}+\sqrt{n}}:\left(\dfrac{m+n}{\sqrt{mn}}+\dfrac{n}{m-\sqrt{mn}}-\dfrac{m}{n+\sqrt{mn}}\right)\)
\(=\dfrac{m+n}{\sqrt{m}+\sqrt{n}}:\left(\dfrac{m+n}{\sqrt{mn}}+\dfrac{n}{\sqrt{m}\left(\sqrt{m}-\sqrt{n}\right)}-\dfrac{m}{\sqrt{n}\left(\sqrt{m}+\sqrt{n}\right)}\right)\)
\(=\dfrac{m+n}{\sqrt{m}+\sqrt{n}}:\left[\dfrac{\left(m+n\right)\left(m-n\right)}{\sqrt{mn}\left(m-n\right)}+\dfrac{n\sqrt{n}\left(\sqrt{m}+\sqrt{n}\right)}{\sqrt{mn}\left(m-n\right)}-\dfrac{m\sqrt{m}\left(\sqrt{m}-\sqrt{n}\right)}{\sqrt{mn}\left(m-n\right)}\right]\)
\(=\dfrac{m+n}{\sqrt{m}+\sqrt{n}}:\dfrac{m^2-n^2+n\sqrt{mn}+n^2-m^2+m\sqrt{mn}}{\sqrt{mn}\left(m-n\right)}\)
\(=\dfrac{m+n}{\sqrt{m}+\sqrt{n}}:\dfrac{n\sqrt{mn}+m\sqrt{mn}}{\sqrt{mn}\left(m-n\right)}\)
\(=\dfrac{m+n}{\sqrt{m}+\sqrt{n}}\cdot\dfrac{\sqrt{mn}\left(\sqrt{m}-\sqrt{n}\right)\left(\sqrt{m}+\sqrt{n}\right)}{\sqrt{mn}\left(m+n\right)}\)
\(=\sqrt{m}-\sqrt{n}\)
