The expression provided in the image is:

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

To simplify this expression, let's go through the steps:

Step 1: Simplify the Radicals

First, notice that $\sqrt{3x^2}$ can be simplified:

$\sqrt{3x^2} = \sqrt{3} \cdot \sqrt{x^2} = \sqrt{3} \cdot x$

So the denominator becomes:

$3\sqrt{3} \cdot x$

Step 2: Write the Expression with the Simplified Denominator

Substituting back into the original expression:

$\frac{4x^3 - 3\sqrt{3x}}{3\sqrt{3} \cdot x}$

Step 3: Simplify the Numerator

The numerator is $4x^3 - 3\sqrt{3x}$, and there's no further simplification possible for the radicals here, so let's divide each term in the numerator by the denominator:

$\frac{4x^3}{3\sqrt{3} \cdot x} - \frac{3\sqrt{3x}}{3\sqrt{3} \cdot x}$

Step 4: Simplify Each Term

Now, simplify each term separately:

1. For $\frac{4x^3}{3\sqrt{3} \cdot x}$:

$\frac{4x^3}{3\sqrt{3} \cdot x} = \frac{4x^2}{3\sqrt{3}}$

2. For $\frac{3\sqrt{3x}}{3\sqrt{3} \cdot x}$:

$\frac{3\sqrt{3x}}{3\sqrt{3} \cdot x} = \frac{\sqrt{3x}}{\sqrt{3} \cdot x} = \frac{\sqrt{3} \cdot \sqrt{x}}{\sqrt{3} \cdot x} = \frac{1}{\sqrt{x}} = \frac{1}{x^{1/2}} = x^{-1/2}$

Final Simplified Expression

Thus, the simplified expression is:

$\frac{4x^2}{3\sqrt{3}} - x^{-1/2}$

Would you like more details on any of these steps? Also, here are some related questions you might find useful:

1. How do you rationalize a denominator involving a square root?
2. What are the properties of exponents used in simplifying radical expressions?
3. How do you factor expressions involving square