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In this image, you have two sets of expressions on the left-hand side and matching options on the right-hand side. The goal seems to be matching the simplified forms of the square root expressions on the left with the correct simplified forms on the right.

Let's break down the expressions on the left and simplify them one by one, then match them to the right-hand side.

1. **Expression E:** \(\sqrt{16x^2y}\)
   - Simplify: 
     \[
     \sqrt{16x^2y} = \sqrt{16} \cdot \sqrt{x^2} \cdot \sqrt{y} = 4x\sqrt{y}
     \]
   - This matches **option E** on the right: \( 4x\sqrt{y} \).

2. **Expression B:** \(\sqrt{4x^2y}\)
   - Simplify:
     \[
     \sqrt{4x^2y} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y} = 2x\sqrt{y}
     \]
   - This matches **option B** on the right: \( 2x\sqrt{y} \).

3. **Expression D:** \(\sqrt{8x^2y}\)
   - Simplify:
     \[
     \sqrt{8x^2y} = \sqrt{4 \cdot 2x^2y} = \sqrt{4} \cdot \sqrt{2x^2y} = 2x\sqrt{2y}
     \]
   - This matches **option A** on the right: \( 2x\sqrt{2y} \).

4. **Expression C:** \(\sqrt{4xy^2}\)
   - Simplify:
     \[
     \sqrt{4xy^2} = \sqrt{4} \cdot \sqrt{xy^2} = 2y\sqrt{x}
     \]
   - This matches **option C** on the right: \( 2y\sqrt{x} \).

5. **Expression F:** \(\sqrt{4x^2y^4}\)
   - Simplify:
     \[
     \sqrt{4x^2y^4} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y^4} = 2x \cdot y^2
     \]
   - This matches **option F** on the right: \( 2xy^2 \).

So, the correct matches are:
- E → \( 4x\sqrt{y} \) (option E)
- B → \( 2x\sqrt{y} \) (option B)
- D → \( 2x\sqrt{2y} \) (option A)
- C → \( 2y\sqrt{x} \) (option C)
- F → \( 2xy^2 \) (option F)

Would you like more details on any of these steps? Here are some related questions to expand your understanding:

1. How would you simplify a general expression involving square roots, such as \(\sqrt{a \cdot b}\)?
2. How does the square root of a product like \( \sqrt{x^2 y} \) simplify in terms of each variable?
3. What are the rules for simplifying powers under square roots, especially for terms like \( y^4 \)?
4. Can these techniques be applied to cube roots or other higher-order roots?
5. How does multiplying or dividing terms inside the square root affect the simplification?

**Tip:** When simplifying square roots, always look for perfect squares within the expression, as they can be easily pulled out of the square root.

Simplify the square root expressions and match them with their corresponding options.

Explore how to simplify square root expressions involving variables like √(16x^2y) and √(4x^2y). This problem covers key algebraic concepts such as the product property of square roots and laws of exponents. Step-by-step solutions are provided for each expression, with matching options included.

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