The question asks to express $\frac{2x^2}{\sqrt{3x^7}}$ in its simplest radical form with a rational denominator.

Let's break it down step by step:

Step 1: Simplify the denominator

The denominator is $\sqrt{3x^7}$. We can rewrite it as: $\sqrt{3x^7} = \sqrt{3} \cdot \sqrt{x^7} = \sqrt{3} \cdot x^{7/2}$

Step 2: Simplify the entire expression

Now, the expression becomes: $\frac{2x^2}{\sqrt{3} \cdot x^{7/2}}$ We can simplify the powers of $x$: $\frac{x^2}{x^{7/2}} = x^{2 - 7/2} = x^{-3/2}$ So the expression becomes: $\frac{2x^{-3/2}}{\sqrt{3}} = \frac{2}{\sqrt{3}} \cdot x^{-3/2}$

Step 3: Rationalize the denominator

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$: $\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$ Thus, the simplified form is: $\frac{2\sqrt{3}}{3} \cdot x^{-3/2}$

This is the simplest radical form.

Final Answer:

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

Would you like further details on any step or have any additional questions? Here are some related questions to explore:

1. How do you simplify radical expressions with variables?
2. What are the steps to rationalize a denominator?
3. How do you handle fractional exponents during simplification?
4. Can this process be applied to more complex radicals?
5. How does rationalizing a denominator affect the value of the expression?

Tip: Always check for opportunities to combine like terms when simplifying expressions with exponents!