To solve this expression, let's break down each part of it systematically.

The expression given is: [ \sqrt[3]{9} \cdot \sqrt[3]{54} \cdot \left( \sqrt[6]{3} \right)^{12} ]

1. Simplify each component separately:

   • First term: $\sqrt[3]{9}$
     Since $9 = 3^2$, we have: [ \sqrt[3]{9} = \sqrt[3]{3^2} = 3^{\frac{2}{3}} ]

   • Second term: $\sqrt[3]{54}$
     Since $54 = 2 \cdot 3^3$, we can write: [ \sqrt[3]{54} = \sqrt[3]{2 \cdot 3^3} = 2^{\frac{1}{3}} \cdot 3 ]

   • Third term: $\left( \sqrt[6]{3} \right)^{12}$
     Here, $\sqrt[6]{3} = 3^{\frac{1}{6}}$, and raising this to the power of 12 gives: $\left( 3^{\frac{1}{6}} \right)^{12} = 3^{2} = 9$

2. Combine all the components:

   Substitute each simplified part back into the expression: $3^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} \cdot 3 \cdot 9$

3. Further simplify the powers of 3:

   Combine the terms with base 3: $3^{\frac{2}{3} + 1 + 2} \cdot 2^{\frac{1}{3}}$

   Calculating the exponent for 3: $\frac{2}{3} + 1 + 2 = \frac{2}{3} + \frac{3}{3} + \frac{6}{3} = \frac{11}{3}$

   So we have: $3^{\frac{11}{3}} \cdot 2^{\frac{1}{3}}$

4. Final Answer: The simplified form of the expression is: $3^{\frac{11}{3}} \cdot 2^{\frac{1}{3}}$

Would you like further details on any of the steps, or do you have any specific questions?

Here are some related questions you might find useful:

1. How do fractional exponents relate to roots?
2. What are the rules for multiplying expressions with the same base?
3. How do you simplify expressions with multiple roots?
4. How can the properties of exponents be used in complex expressions?
5. What methods exist for evaluating cube roots of non-perfect cubes?

Tip: When simplifying roots, try to express numbers in terms of their prime factors to reveal simplifications easily.