What is \( \frac{\sqrt{12 x^{8}}}{\sqrt{3 x^{2}}} \) in simplest form, where \( x \geq 0 ? \)

To simplify \( \frac{\sqrt{12 x^{8}}}{\sqrt{3 x^{2}}} \), we can first combine the square roots. This gives us: \[ \sqrt{\frac{12 x^{8}}{3 x^{2}}} \] Now, simplifying the fraction inside the square root: \[ \frac{12}{3} = 4 \quad \text{and} \quad \frac{x^{8}}{x^{2}} = x^{8-2} = x^{6} \] So we have: \[ \sqrt{4 x^{6}} = \sqrt{4} \cdot \sqrt{x^{6}} = 2 \cdot x^{3} \] Thus, the expression simplifies to: \[ 2x^{3} \] Therefore, the simplest form of \( \frac{\sqrt{12 x^{8}}}{\sqrt{3 x^{2}}} \) is: \[ 2x^{3} \]