What is the following quotient? \( \frac{3 \sqrt{8}}{4 \sqrt{6}} \)

To simplify the quotient \( \frac{3 \sqrt{8}}{4 \sqrt{6}} \), we start by simplifying \( \sqrt{8} \). Since \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \), the expression becomes: \[ \frac{3 \cdot 2\sqrt{2}}{4 \sqrt{6}} = \frac{6\sqrt{2}}{4\sqrt{6}} \] Next, we can simplify the fraction by dividing both the numerator and the denominator by 2: \[ \frac{6\sqrt{2}}{4\sqrt{6}} = \frac{3\sqrt{2}}{2\sqrt{6}} \] Now, to rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{6} \): \[ \frac{3\sqrt{2} \cdot \sqrt{6}}{2\sqrt{6} \cdot \sqrt{6}} = \frac{3\sqrt{12}}{2 \cdot 6} = \frac{3\sqrt{12}}{12} \] Simplifying \( \sqrt{12} = 2\sqrt{3} \) gives: \[ \frac{3 \cdot 2\sqrt{3}}{12} = \frac{6\sqrt{3}}{12} = \frac{\sqrt{3}}{2} \] Thus, the final simplified quotient is: \[ \frac{\sqrt{3}}{2} \]