\(\begin{array}{l} Q = \left( {\frac{1}{{\sqrt a + 1}} - \frac{1}{{a + \sqrt a }}} \right):\frac{{\sqrt a - 1}}{{a + 2\sqrt a + 1}}\\ = \left[ {\frac{1}{{\sqrt a + 1}} - \frac{1}{{\sqrt a \left( {\sqrt a + 1} \right)}}} \right]:\frac{{\sqrt a - 1}}{{{{\left( {\sqrt a + 1} \right)}^2}}}\\ = \frac{{\sqrt a - 1}}{{\sqrt a \left( {\sqrt a + 1} \right)}}.\frac{{{{\left( {\sqrt a + 1} \right)}^2}}}{{\sqrt a - 1}}\\ = \frac{{\sqrt a + 1}}{{\sqrt a }} \end{array}\)
* Xét hiệu
\(\begin{array}{l} Q - 1 = \frac{{\sqrt a + 1}}{{\sqrt a }} - 1\\ = \frac{{\sqrt a + 1 - \sqrt a }}{{\sqrt a }} = \frac{1}{{\sqrt a }} > 0\,\,\left( {a > 0} \right)\\ \Rightarrow Q - 1 > 0 \Rightarrow Q > 1 \end{array}\)