Divide these numbers using fractional exponents. \( \frac{\sqrt{6^{3}}}{\sqrt[4]{6}} \)

To divide the numbers using fractional exponents, we can start by expressing the square root and the fourth root in terms of exponents. The expression \(\sqrt{6^{3}}\) can be written as \(6^{3/2}\), and \(\sqrt[4]{6}\) can be expressed as \(6^{1/4}\). Now, we can rewrite the original expression: \[ \frac{\sqrt{6^{3}}}{\sqrt[4]{6}} = \frac{6^{3/2}}{6^{1/4}} \] When dividing like bases, we subtract the exponents: \[ 6^{3/2 - 1/4} \] To subtract the exponents, we need a common denominator. The least common denominator between 2 and 4 is 4. Thus, we convert \(3/2\) as follows: \[ 3/2 = \frac{3 \times 2}{2 \times 2} = \frac{6}{4} \] Now we can substitute back into our exponent subtraction: \[ 6^{\frac{6}{4} - \frac{1}{4}} = 6^{\frac{5}{4}} \] Thus, the final answer is: \[ 6^{\frac{5}{4}} \]