\[\frac{a}{a - 1} - \frac{\sqrt{a}}{\sqrt{a} + 1} =\]
\[= \frac{a}{\left( \sqrt{a} - 1 \right)\left( \sqrt{a} + 1 \right)} - \frac{{\sqrt{a}}^{\backslash\text{√}a - 1}}{\sqrt{a} + 1} =\]
\[= \frac{a - a + \sqrt{a}}{a - 1} = \frac{\sqrt{a}}{a - 1}.\]