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To simplify \( \sqrt{75} \) using the product rule, we can look for factors of 75 that include a perfect square.

1. **Factorize 75**:  
   \( 75 = 25 \times 3 \)

2. **Apply the product rule** for square roots, which states \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \):  
   \[
   \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}
   \]

3. **Simplify** each square root:  
   \[
   \sqrt{25} = 5 \quad \text{and} \quad \sqrt{3} \text{ remains as } \sqrt{3}
   \]

4. **Combine the terms**:  
   \[
   \sqrt{75} = 5\sqrt{3}
   \]

So, the simplified form of \( \sqrt{75} \) is:
\[
5\sqrt{3}
\]

Would you like more details on this process or have any further questions?

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Here are five related questions that expand on this concept:

1. How would you simplify a radical with larger numbers, like \( \sqrt{300} \)?
2. What is the product rule for radicals, and how is it generally applied?
3. How can you identify when a number under a radical has a perfect square factor?
4. How do you handle simplification for cube roots or other higher roots?
5. Can the product rule for radicals be used in reverse to combine separate square roots?

**Tip:** When simplifying radicals, always look for the largest perfect square factor to simplify the process efficiently.

Use the product rule to simplify the radical √75.

Learn how to simplify the radical √75 using the product rule for square roots. This step-by-step solution explores factoring, applying the product rule, and simplifying square roots. Ideal for students in Grades 7-9.

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