Rationalize the denominator and simplify: \( \frac{\sqrt{6 x^{3}+\sqrt{5 x^{5}}}}{\sqrt{3 x^{2}}} \)

To rationalize the denominator and simplify the expression \( \frac{\sqrt{6 x^{3}+\sqrt{5 x^{5}}}}{\sqrt{3 x^{2}}} \), we start by simplifying both the numerator and the denominator. The denominator is: \[ \sqrt{3x^{2}} = \sqrt{3} \cdot \sqrt{x^{2}} = \sqrt{3} \cdot x \] Now let's focus on the numerator \( \sqrt{6 x^{3}+\sqrt{5 x^{5}}} \): We can simplify \( \sqrt{5 x^{5}} \) as follows: \[ \sqrt{5 x^{5}} = \sqrt{5} \cdot \sqrt{x^{5}} = \sqrt{5} \cdot x^{\frac{5}{2}} = \sqrt{5} \cdot x^2 \cdot \sqrt{x} = x^2 \sqrt{5x} \] Thus, the numerator becomes: \[ \sqrt{6 x^{3} + x^2 \sqrt{5x}} = \sqrt{x^2(6x + \sqrt{5x})} = x \sqrt{6x + \sqrt{5x}} \] Putting it all together: \[ \frac{\sqrt{6 x^{3} + \sqrt{5 x^{5}}}}{\sqrt{3 x^{2}}} = \frac{x \sqrt{6x + \sqrt{5x}}}{\sqrt{3} \cdot x} \] The \( x \) terms in the numerator and denominator cancel (assuming \( x \neq 0 \)): \[ = \frac{\sqrt{6x + \sqrt{5x}}}{\sqrt{3}} = \frac{1}{\sqrt{3}} \sqrt{6x + \sqrt{5x}} \] Finally, we can simplify further: \[ = \frac{\sqrt{6x + \sqrt{5x}}}{\sqrt{3}} = \sqrt{\frac{6x + \sqrt{5x}}{3}} \] Thus, the simplified expression is: \[ \sqrt{\frac{6x + \sqrt{5x}}{3}} \]