The expression in the image is:

$\frac{3\sqrt{12}}{4 \sqrt[3]{54}}$

Step-by-step simplification:

1. Simplify the square root of 12: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ Therefore, $3\sqrt{12} = 3 \times 2\sqrt{3} = 6\sqrt{3}$.

2. Simplify the cube root of 54: [ \sqrt[3]{54} = \sqrt[3]{2 \times 27} = \sqrt[3]{27 \times 2} = 3 \times \sqrt[3]{2} ] So, $4 \sqrt[3]{54} = 4 \times 3 \times \sqrt[3]{2} = 12 \sqrt[3]{2}$.

3. Combine the expression: [ \frac{6\sqrt{3}}{12\sqrt[3]{2}} = \frac{6}{12} \times \frac{\sqrt{3}}{\sqrt[3]{2}} = \frac{1}{2} \times \frac{\sqrt{3}}{\sqrt[3]{2}} ]

4. Final result: $\frac{\sqrt{3}}{2\sqrt[3]{2}}$

This is the simplified version of the expression.

Let me know if you'd like any further details or if you have questions!

Here are five related questions:

1. How do you rationalize an expression with cube roots?
2. What are the properties of cube roots compared to square roots?
3. Can you express this result in a decimal form for approximation?
4. How would this change if the numerator had a cube root instead of a square root?
5. Can this expression be further simplified using logarithms?

Tip: Always check if numbers under a square root or cube root can be factored into smaller prime components for easier simplification.