\( ( \sqrt { \frac { 4 a - 4 } { a ^ { 2 } } } + \sqrt { a ^ { 3 } - a ^ { 2 } } + \sqrt { 9 a - 9 } ) : ( 1 + \frac { 2 } { a } ) - \sqrt { a - 1 } \)

To simplify the expression \( ( \sqrt { \frac { 4 a - 4 } { a ^ { 2 } } } + \sqrt { a ^ { 3 } - a ^ { 2 } } + \sqrt { 9 a - 9 } ) : ( 1 + \frac { 2 } { a } ) - \sqrt { a - 1 } \), let's break it down step by step. 1. **First Term: \( \sqrt { \frac { 4a - 4 } { a^{2} } } \)** Simplifying inside the square root: \[ \frac{4a - 4}{a^{2}} = \frac{4(a - 1)}{a^{2}} = \frac{4}{a} - \frac{4}{a^{2}} \] Therefore, \[ \sqrt{\frac{4(a - 1)}{a^{2}}} = \frac{2\sqrt{a - 1}}{a} \] 2. **Second Term: \( \sqrt { a^{3} - a^{2} } \)** Factoring out \( a^{2} \): \[ \sqrt{a^{2}(a - 1)} = a\sqrt{a - 1} \] 3. **Third Term: \( \sqrt { 9a - 9 } \)** Factoring out \( 9 \): \[ \sqrt{9(a - 1)} = 3\sqrt{a - 1} \] Putting this all together, we have: \[ \sqrt { \frac { 4a - 4 } { a^{2} } } + \sqrt { a^{3} - a^{2} } + \sqrt { 9a - 9 } = \frac{2\sqrt{a - 1}}{a} + a\sqrt{a - 1} + 3\sqrt{a - 1} \] Combining like terms: \[ = \sqrt{a - 1}\left(\frac{2}{a} + a + 3\right) \] Putting this back into the expression: \[ \left(\sqrt{a - 1}\left(\frac{2}{a} + a + 3\right)\right) : \left(1 + \frac{2}{a}\right) - \sqrt{a - 1} \] 4. **Dividing by \( 1 + \frac{2}{a} = \frac{a + 2}{a} \)**: This division can be represented as multiplying by the reciprocal: \[ \text{Dividing} : \sqrt{a - 1}\left(\frac{2 + 3a + a^2}{a}\right) \times \left(\frac{a}{a + 2}\right) \] This simplifies to: \[ \sqrt{a - 1} \frac{2 + 3a + a^{2}}{a + 2} \] Putting it all together, the final expression becomes: \[ \sqrt{a - 1} \left(\frac{2 + 3a + a^{2}}{a + 2} - 1\right) \] 5. **Combine the terms**: This final step combines the two parts \( \left(\frac{2 + 3a + a^{2}}{a + 2}\right) - 1 \). The complete expression will depend on \( a - 1 \) being defined (i.e. \( a \geq 1 \)). Thus the cleaned-up version of the expression appears as: \[ \sqrt{a - 1} \cdot \left(\frac{2 + 3a + a^2 - (a + 2)}{a + 2}\right) = \sqrt{a - 1} \cdot \left(\frac{(a^2 + 2a)}{a + 2}\right) \] So the entire expression can be summarized. As a final answer: \[ \frac{(a^2 + 2a)}{a + 2} \cdot \sqrt{a - 1} \]