To simplify the given expression:

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

Let's follow these steps:

Step 1: Simplify the Denominator

The denominator is $3\sqrt[3]{3x^2}$.

Step 2: Break Down the Expression

To make the expression simpler, let's consider each term separately.

1. Numerator: $4x^3 - 3\sqrt{3x}$.
2. Denominator: $3\sqrt[3]{3x^2}$.

Step 3: Simplify Each Part

Numerator:

The numerator has two terms:

• $4x^3$ is already simplified.
• For $-3\sqrt{3x}$, it is already in its simplest form.

Denominator:

For the denominator, we have:

• $3\sqrt[3]{3x^2}$, which is $3 \cdot (3x^2)^{1/3}$.

Now let's rewrite the expression without simplifying further due to lack of common factors.

Step 4: Combine

We combine both the numerator and the denominator:

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

There is no further simplification since no common factors exist between the numerator and the denominator that we can factor out.

Would you like more details or to explore a different aspect of this problem? Here are some related questions:

1. How do you factor expressions with radicals?
2. Can we rationalize the denominator further if more factors were present?
3. How does simplifying change if values for $x$ are provided?
4. How to simplify expressions with both square and cube roots combined?
5. What are the steps to simplify complex fractions?

Tip: Always look for common factors to factor out, as this is the key to simplifying many expressions!