We are asked to find the value of the expression $\frac{6^{1/2} \cdot 9^{1/4}}{216^{1/4}}$.

1. Problem Description

We are asked to find the value of the expression $\frac{6^{1/2} \cdot 9^{1/4}}{216^{1/4}}$.

2. Solution Steps

We simplify the expression:

$\frac{6^{1/2} \cdot 9^{1/4}}{216^{1/4}} = \frac{6^{1/2} \cdot (3^2)^{1/4}}{(6^3)^{1/4}} = \frac{6^{1/2} \cdot 3^{2/4}}{6^{3/4}} = \frac{6^{1/2} \cdot 3^{1/2}}{6^{3/4}} = \frac{(6 \cdot 3)^{1/2}}{6^{3/4}} = \frac{18^{1/2}}{6^{3/4}} = \frac{(2 \cdot 3^2)^{1/2}}{(2 \cdot 3)^{3/4}} = \frac{2^{1/2} \cdot 3^{2/2}}{2^{3/4} \cdot 3^{3/4}} = \frac{2^{1/2} \cdot 3}{2^{3/4} \cdot 3^{3/4}} = 2^{\frac{1}{2} - \frac{3}{4}} \cdot 3^{1 - \frac{3}{4}} = 2^{\frac{2}{4} - \frac{3}{4}} \cdot 3^{\frac{4}{4} - \frac{3}{4}} = 2^{-1/4} \cdot 3^{1/4} = \frac{3^{1/4}}{2^{1/4}} = (\frac{3}{2})^{1/4}$

3. Final Answer

(c) $(\frac{3}{2})^{1/4}$