https://file.aiquickdraw.com/mathbot/file/1726590898351oy0edksd.jpg

The problem in the image asks for the product of \( \sqrt{2x^2} \) and \( \sqrt{4x^2y^6} \) in its simplest form, and it provides four answer options.

Let’s break this down step by step:

1. **Simplify each square root separately:**

   - \( \sqrt{2x^2} = \sqrt{2} \cdot \sqrt{x^2} = \sqrt{2} \cdot x \)
   - \( \sqrt{4x^2y^6} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y^6} = 2x \cdot y^3 \)

2. **Now, multiply the simplified forms:**

   \( (\sqrt{2} \cdot x) \times (2x \cdot y^3) = \sqrt{2} \cdot 2 \cdot x \cdot x \cdot y^3 = 2\sqrt{2} \cdot x^2 \cdot y^3 \)

So the product of \( \sqrt{2x^2} \) and \( \sqrt{4x^2y^6} \) in its simplest form is \( 2\sqrt{2} \cdot x^2 \cdot y^3 \).

Let's compare this result with the given answer choices:

- (a) \( \sqrt{8x^3y^6} \)
- (b) \( \sqrt{2xy^3} \)
- (c) \( \sqrt{2x^2y^6} \)
- (d) \( 2xy^2 \)

None of these options exactly match the simplified form \( 2\sqrt{2} \cdot x^2 \cdot y^3 \), but the closest expression that resembles our answer is **(b) \( \sqrt{2xy^3} \)**, though simplified differently.

### Do you have any questions or would you like further details?

Here are 5 related questions:
1. How do you simplify products of square roots?
2. What are the rules for simplifying square roots involving variables?
3. How can you multiply square roots with coefficients?
4. How do you identify perfect squares when simplifying square roots?
5. How does multiplication of terms inside a square root affect the final result?

**Tip:** Always break down complex square roots by simplifying each part, especially when variables are involved!

The product of \( \sqrt{2x^2} \) and \( \sqrt{4x^2y^6} \) in simplest form is:

This problem simplifies the product of two square roots, \( \sqrt{2x^2} \) and \( \sqrt{4x^2y^6} \), using square root properties and exponent rules. It is suitable for students in Grades 9-10.

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