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Let's solve the expression \(\sqrt{147x^2}\) using the product rule for radicals.

1. Break the expression inside the square root into factors:
   \[
   \sqrt{147x^2} = \sqrt{147} \cdot \sqrt{x^2}
   \]

2. Simplify \(\sqrt{x^2}\):
   \[
   \sqrt{x^2} = x
   \]

3. Now, simplify \(\sqrt{147}\). We find the prime factorization of 147:
   \[
   147 = 3 \cdot 49 = 3 \cdot (7^2)
   \]

   So:
   \[
   \sqrt{147} = \sqrt{3 \cdot 7^2} = 7\sqrt{3}
   \]

4. Putting it all together:
   \[
   \sqrt{147x^2} = 7\sqrt{3} \cdot x = 7x\sqrt{3}
   \]

Thus, the simplified expression is:
\[
7x\sqrt{3}
\]

Let me know if you'd like further explanation or have any questions!

### Related Questions:
1. How do you simplify the square root of a product involving variables?
2. What is the prime factorization method for simplifying square roots?
3. How do you handle square roots of perfect squares in expressions?
4. How does the product rule for radicals apply to more complex expressions?
5. Can you apply this method to cube roots or higher-order roots?

### Tip:
When simplifying radicals with variables, always check if the exponent on the variable allows for perfect squares to be factored out.

Use the product rule to simplify the following expression: \(\sqrt{147x^2}\).

Learn how to simplify the radical expression \(\sqrt{147x^2}\) using the product rule and prime factorization. This example breaks down the steps to simplify square roots, making it easy for students in Grades 8-10 to follow.

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