Rút gọn C = \left( \frac{\sqrt{x}}{\sqrt{x-1}} + \frac{2}{x-\sqrt{x}} \right) \cdot \frac{1}{\sqrt{x-1}}

To simplify the expression

\[
C = \left( \frac{\sqrt{x}}{\sqrt{x-1}} + \frac{2}{x-\sqrt{x}} \right) \cdot \frac{1}{\sqrt{x-1}},
\]

follow these steps:

1. **Rewrite the second term**: Start with the second term \( \frac{2}{x - \sqrt{x}} \). We can simplify \( x - \sqrt{x} \):
\[
x - \sqrt{x} = \sqrt{x}(\sqrt{x} - 1),
\]
so,
\[
\frac{2}{x - \sqrt{x}} = \frac{2}{\sqrt{x}(\sqrt{x} - 1)} = \frac{2}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}-1}.
\]

2. **Combine the terms inside the parentheses**:
\[
\frac{\sqrt{x}}{\sqrt{x-1}} + \frac{2}{\sqrt{x}(\sqrt{x}-1)}.
\]

To add these two fractions, find a common denominator, which is \( \sqrt{x-1} \cdot \sqrt{x}(\sqrt{x} - 1) \):
\[
= \frac{\sqrt{x}(\sqrt{x}(\sqrt{x}-1)) + 2\sqrt{x - 1}}{\sqrt{x-1}\cdot \sqrt{x}(\sqrt{x}-1)}.
\]

3. **Evaluate the entire expression**:
\[
C = \left( \frac{\sqrt{x} \cdot (\sqrt{x}(\sqrt{x}-1)) + 2\sqrt{x-1}}{\sqrt{x-1}\cdot \sqrt{x}(\sqrt{x}-1)} \right) \cdot \frac{1}{\sqrt{x-1}}.
\]
This simplifies further:
\[
C = \frac{\sqrt{x}(\sqrt{x}(\sqrt{x}-1)) + 2\sqrt{x-1}}{(\sqrt{x-1})^2 \cdot \sqrt{x}(\sqrt{x}-1)}.
\]

4. **Final Form**:
This will yield the final simplified form of \( C \), combining terms wherever necessary.

Thus, the simplified expression for \( C \) is:

\[
C = \frac{\sqrt{x}(\sqrt{x}(\sqrt{x}-1)) + 2\sqrt{x-1}}{(x-1) \sqrt{x}(\sqrt{x}-1)}.
\]