Perform the operations and simplify: \( \frac{\sqrt{3 x^{3}}}{\sqrt{5 x}} \) \( \sqrt{9 x^{6}} \) \( 9 x^{6} \) \( 3 x^{3} \) \( 3 x^{2} \sqrt{x^{3}} \)

To simplify the expression \( \frac{\sqrt{3 x^{3}}}{\sqrt{5 x}} \), we can start by using the property of square roots that states \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). So, we have: \[ \frac{\sqrt{3 x^{3}}}{\sqrt{5 x}} = \sqrt{\frac{3 x^{3}}{5 x}} \] Next, we simplify the fraction inside the square root: \[ \frac{3 x^{3}}{5 x} = \frac{3}{5} \cdot \frac{x^{3}}{x} = \frac{3}{5} \cdot x^{2} \] Now, substituting this back into the square root gives us: \[ \sqrt{\frac{3}{5} \cdot x^{2}} = \sqrt{\frac{3}{5}} \cdot \sqrt{x^{2}} = \sqrt{\frac{3}{5}} \cdot x \] Thus, the simplified form of \( \frac{\sqrt{3 x^{3}}}{\sqrt{5 x}} \) is: \[ x \sqrt{\frac{3}{5}} \] Now, let's evaluate the other expressions one by one: 1. **\( \sqrt{9 x^{6}} \)**: \[ \sqrt{9 x^{6}} = \sqrt{9} \cdot \sqrt{x^{6}} = 3 \cdot x^{3} = 3 x^{3} \] 2. **\( 9 x^{6} \)**: This expression is already simplified as \( 9 x^{6} \). 3. **\( 3 x^{3} \)**: This expression is already simplified as \( 3 x^{3} \). 4. **\( 3 x^{2} \sqrt{x^{3}} \)**: We can simplify \( \sqrt{x^{3}} \) as follows: \[ \sqrt{x^{3}} = \sqrt{x^{2} \cdot x} = \sqrt{x^{2}} \cdot \sqrt{x} = x \sqrt{x} \] Therefore, \[ 3 x^{2} \sqrt{x^{3}} = 3 x^{2} \cdot x \sqrt{x} = 3 x^{3} \sqrt{x} \] In summary, the simplified results are: - \( \frac{\sqrt{3 x^{3}}}{\sqrt{5 x}} = x \sqrt{\frac{3}{5}} \) - \( \sqrt{9 x^{6}} = 3 x^{3} \) - \( 9 x^{6} \) (already simplified) - \( 3 x^{3} \) (already simplified) - \( 3 x^{2} \sqrt{x^{3}} = 3 x^{3} \sqrt{x} \)