https://file.aiquickdraw.com/mathbot/file/17281683521770ii8827g.jpg

The expression shown in the image is:

\[
\frac{30\sqrt{5}}{10\sqrt{2}}
\]

Let's simplify this step by step:

1. **Simplify the coefficients**:  
   The fraction involving the numbers \( \frac{30}{10} \) simplifies to 3.
   So, we have:
   \[
   3 \cdot \frac{\sqrt{5}}{\sqrt{2}}
   \]

2. **Simplify the square roots**:  
   We can combine the square roots into one:
   \[
   3 \cdot \sqrt{\frac{5}{2}}
   \]

3. **Final expression**:  
   The simplified expression is:
   \[
   3\sqrt{\frac{5}{2}}
   \]

This is the simplest form of the given expression.

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions to expand on this:

1. How would you simplify \( \frac{\sqrt{a}}{\sqrt{b}} \) in general?
2. Can you apply this technique to simplify \( \frac{50\sqrt{7}}{25\sqrt{3}} \)?
3. What is the geometric interpretation of multiplying or dividing square roots?
4. How do we rationalize a denominator if a square root is in it?
5. How does the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \) help in simplifying expressions like this?

**Tip**: When simplifying expressions involving square roots, always check if the coefficients (numbers outside the square roots) can be simplified first to make the calculations easier.

Simplify the expression \( \frac{30\sqrt{5}}{10\sqrt{2}} \)

Learn how to simplify the radical expression \( \frac{30\sqrt{5}}{10\sqrt{2}} \) by breaking it down step by step. This problem involves simplifying coefficients and combining square roots using algebraic principles. Perfect for students in Grades 9-10.

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