The problem asks to find the value of the expression $\frac{6^{1/4} 294^{1/4}}{216^{1/4}}$.

1. Problem Description

The problem asks to find the value of the expression $\frac{6^{1/4} 294^{1/4}}{216^{1/4}}$.

2. Solution Steps

We want to simplify the expression $\frac{6^{1/4} 294^{1/4}}{216^{1/4}}$.

First, we can combine the terms in the numerator:

$6^{1/4} \cdot 294^{1/4} = (6 \cdot 294)^{1/4}$.

Now, let's calculate $6 \cdot 294 = 6 \cdot (6 \cdot 49) = 36 \cdot 49 = 6^2 \cdot 7^2 = (6 \cdot 7)^2 = 42^2 = 1764$.

So, $6^{1/4} \cdot 294^{1/4} = (1764)^{1/4} = (42^2)^{1/4} = 42^{2/4} = 42^{1/2} = \sqrt{42}$.

Next, we need to find the value of $216^{1/4}$.

Note that $216 = 6^3$. Thus, $216^{1/4} = (6^3)^{1/4} = 6^{3/4}$.

Therefore, the expression becomes:

$\frac{6^{1/4} 294^{1/4}}{216^{1/4}} = \frac{(6 \cdot 294)^{1/4}}{(6^3)^{1/4}} = \frac{1764^{1/4}}{6^{3/4}} = \frac{(42^2)^{1/4}}{6^{3/4}} = \frac{42^{1/2}}{6^{3/4}} = \frac{(6 \cdot 7)^{1/2}}{6^{3/4}} = \frac{6^{1/2} 7^{1/2}}{6^{3/4}} = 6^{1/2 - 3/4} 7^{1/2} = 6^{-1/4} 7^{1/2} = \frac{7^{1/2}}{6^{1/4}} = \frac{\sqrt{7}}{\sqrt[4]{6}}$

This is not one of the answer options. Let's re-evaluate.

We have $\frac{6^{1/4} 294^{1/4}}{216^{1/4}} = \left(\frac{6 \cdot 294}{216}\right)^{1/4}$.

$\frac{6 \cdot 294}{216} = \frac{6 \cdot 6 \cdot 49}{6 \cdot 36} = \frac{6 \cdot 6 \cdot 7 \cdot 7}{6 \cdot 6 \cdot 6} = \frac{49}{6}$.

So, we want to find the value of $\left(\frac{49}{6}\right)^{1/4} = \left(\frac{7^2}{6}\right)^{1/4} = \frac{7^{2/4}}{6^{1/4}} = \frac{7^{1/2}}{6^{1/4}} = \frac{\sqrt{7}}{\sqrt[4]{6}}$.

Let's check if $\frac{49}{6}$ is equivalent to any of the other options.

$\frac{6^{1/4} 294^{1/4}}{216^{1/4}} = \left(\frac{6 \cdot 294}{216}\right)^{1/4} = \left(\frac{1764}{216}\right)^{1/4} = \left(\frac{49}{6}\right)^{1/4}$.

However, $294 = 6 \times 49 = 6 \times 7^2$.

So $\frac{6 \cdot 294}{216} = \frac{6 \cdot (6 \cdot 49)}{6^3} = \frac{6^2 \cdot 7^2}{6^3} = \frac{7^2}{6} = \frac{49}{6}$.

$\left(\frac{49}{6}\right)^{1/4} \approx 1.68$.

Let's revisit the provided options:

(a) $\frac{2}{3}^{1/4} \approx 0.90$

(b) $\frac{3}{4}^{1/4} \approx 0.96$

(c) $\frac{3}{2}^{1/4} \approx 1.10$

(d) $\frac{4}{3}^{1/4} \approx 1.07$

$\frac{6^{1/4} \cdot (6 \cdot 49)^{1/4}}{6^{3/4}} = \frac{6^{1/4} \cdot 6^{1/4} \cdot 49^{1/4}}{6^{3/4}} = \frac{6^{2/4} \cdot (7^2)^{1/4}}{6^{3/4}} = \frac{6^{1/2} \cdot 7^{1/2}}{6^{3/4}} = \frac{\sqrt{6} \sqrt{7}}{6^{3/4}} = \frac{\sqrt{42}}{6^{3/4}}$.

$\left(\frac{49}{6}\right)^{1/4} = \left(\frac{7^2}{6}\right)^{1/4} = \frac{7^{1/2}}{6^{1/4}} = \frac{\sqrt{7}}{\sqrt[4]{6}}$.

3. Final Answer

None of the given options are correct. The answer is $\left(\frac{49}{6}\right)^{1/4}$ or $\frac{\sqrt{7}}{\sqrt[4]{6}}$.