We need to simplify the given expression:

$\frac{2\sqrt{2}}{3\sqrt[3]{3x^4}}$

Step-by-step Solution:

1. Understand the components:

   • $2\sqrt{2}$: The numerator has a coefficient of $2$ multiplied by the square root of $2$.
   • $3\sqrt[3]{3x^4}$: The denominator has a coefficient of $3$ multiplied by the cube root of $3x^4$.

2. Express cube root more clearly: The denominator is $3 \cdot \sqrt[3]{3} \cdot \sqrt[3]{x^4}$.

3. Combine the terms: The fraction is rewritten as: [ \frac{2\sqrt{2}}{3 \cdot \sqrt[3]{3} \cdot \sqrt[3]{x^4}} ]

4. Simplify powers where possible:

   • There are no like terms to simplify directly because the square root ($\sqrt{\cdot}$) and cube root ($\sqrt[3]{\cdot}$) do not simplify with each other.

5. Rewrite in simpler notation: The expression is already in its most reduced form: $\frac{2\sqrt{2}}{3\sqrt[3]{3x^4}}$

Final Answer:

$\boxed{\frac{2\sqrt{2}}{3\sqrt[3]{3x^4}}}$

Let me know if you'd like to see another approach or further explanation!

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Relative Questions:

1. How do square roots and cube roots interact in expressions?
2. How can we simplify expressions with both radicals and cube roots?
3. What are the steps to rationalize a denominator with a cube root?
4. How does the order of roots affect simplification (e.g., square root vs cube root)?
5. What happens if $x$ is negative—does the cube root change?

Tip: When working with roots of different orders, try expressing them as fractional exponents for easier manipulation!