To simplify the expression \( \frac{3\sqrt{216}}{27^{1/3}} \), let's break it down step-by-step.
1. **Calculate \(3\sqrt{216}\)**:
- The square root of 216 can be simplified:
\[
216 = 6^3 = (6^2) \cdot 6 = 36 \cdot 6
\]
- This means:
\[
\sqrt{216} = \sqrt{36 \cdot 6} = \sqrt{36} \cdot \sqrt{6} = 6\sqrt{6}
\]
- Therefore, \(3\sqrt{216}\) becomes:
\[
3\sqrt{216} = 3 \cdot 6\sqrt{6} = 18\sqrt{6}
\]
2. **Calculate \(27^{1/3}\)**:
- Since \(27 = 3^3\),
\[
27^{1/3} = (3^3)^{1/3} = 3^{3 \cdot \frac{1}{3}} = 3^1 = 3
\]
3. **Putting it all together**:
- The expression simplifies to:
\[
\frac{18\sqrt{6}}{3} = 6\sqrt{6}
\]
4. **Evaluating the options**:
- The options provided (18, 8, 2, 12) do not directly match \(6\sqrt{6}\), but let's convert \(6\sqrt{6}\) to a numerical value for clarity:
- Approximating \(\sqrt{6} \approx 2.45\):
\[
6\sqrt{6} \approx 6 \cdot 2.45 \approx 14.7
\]
Since \(6\sqrt{6}\) does not fit exactly into any of the response options, the correct answer appears not listed. However, if you were looking only for numerical simplifications, and since \(6\sqrt{6}\) is closest to 12, of the choices given, **12** could be the most suitable answer when approximating.
Therefore, the answer would be: **12**.