Sửa đề: \(M=\frac{\sqrt{a}+\sqrt{b}+1}{a+a\cdot\sqrt{b}}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\left(\frac{\sqrt{b}}{a-\sqrt{ab}}+\frac{\sqrt{b}}{a+\sqrt{ab}}\right)\)
a: ĐKXĐ: a>0; b>0
b: Ta có: \(M=\frac{\sqrt{a}+\sqrt{b}+1}{a+a\cdot\sqrt{b}}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\left(\frac{\sqrt{b}}{a-\sqrt{ab}}+\frac{\sqrt{b}}{a+\sqrt{ab}}\right)\)
\(=\frac{\sqrt{a}+\sqrt{b}+1}{a\left(\sqrt{b}+1\right)}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\left(\frac{\sqrt{b}}{\sqrt{a}\left(\sqrt{a}-\sqrt{b}\right)}+\frac{\sqrt{b}}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}\right)\)
\(=\frac{\sqrt{a}+\sqrt{b}+1}{a\left(\sqrt{b}+1\right)}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\left(\frac{\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}+\frac{\sqrt{b}\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}\right)\)
\(=\frac{\sqrt{a}+\sqrt{b}+1}{a\left(\sqrt{b}+1\right)}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\cdot\frac{\sqrt{ab}+b+\sqrt{ab}-b}{\sqrt{a}\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\)
\(=\frac{\sqrt{a}+\sqrt{b}+1}{a\left(\sqrt{b}+1\right)}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\cdot\frac{2\sqrt{ab}}{\sqrt{a}\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\)
\(=\frac{\sqrt{a}+\sqrt{b}+1}{a\left(\sqrt{b}+1\right)}+\frac{1}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}=\frac{\left(\sqrt{a}+\sqrt{b}+1\right)\left(\sqrt{a}+\sqrt{b}\right)+\sqrt{a}\left(\sqrt{b}+1\right)}{a\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{b}+1\right)}\)
\(=\frac{\left(\sqrt{a}+\sqrt{b}\right)^2+\sqrt{a}+\sqrt{b}+\sqrt{ab}+\sqrt{a}}{a\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{b}+1\right)}=\frac{\left(\sqrt{a}+\sqrt{b}\right)^2+\sqrt{ab}+\sqrt{b}+2\sqrt{a}}{a\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{b}+1\right)}\)
=>M>0
