The expression given is:

\[
\frac{5n^2}{3\sqrt{mn}}
\]

To simplify this expression, we need to break it down step by step:

### Simplification Process:
1. **Identify the components:**
   - The numerator is \(5n^2\).
   - The denominator is \(3\sqrt{mn}\).

2. **Check for simplifications:**
   - The expression does not have any common factors that can be simplified directly between the numerator and the denominator.

### Final Simplified Expression:

The expression is already in its simplest form:
\[
\frac{5n^2}{3\sqrt{mn}}
\]

If you need to rationalize the denominator, you would multiply the numerator and the denominator by \(\sqrt{mn}\):

\[
\frac{5n^2}{3\sqrt{mn}} \times \frac{\sqrt{mn}}{\sqrt{mn}} = \frac{5n^2\sqrt{mn}}{3mn}
\]

But unless specifically required to rationalize the denominator, the initial form is considered simplified.

Would you like to go over the steps in more detail or have any other questions?

### Questions for Further Exploration:
1. How do you multiply and divide expressions involving square roots?
2. Can you explain the process of rationalizing the denominator?
3. How would this expression change if \(n\) was inside the square root?
4. What is the difference between simplifying and rationalizing an expression?
5. How would you factor the numerator or denominator in expressions like this?
6. What happens if \(m\) and \(n\) are both perfect squares?
7. How would you approach simplifying an expression where \(mn\) is under a cube root instead of a square root?
8. Can you provide an example of a similar expression with different variables and simplify it?

**Tip:** Rationalizing the denominator helps in making expressions easier to work with, especially when dealing with more complex algebraic operations later on.

Learn how to simplify the algebraic expression 5n^2 / (3 * sqrt(mn)) step-by-step. Understand the process of fraction simplification involving variables and square roots. Suitable for students in Grades 9-12.

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