https://file.aiquickdraw.com/mathbot/file/1729632362082l1gqd3nd.jpg

Let's simplify the expression:

\[
\sqrt{80x^4y^3}
\]

### Step 1: Break down the square root

We can break down the square root as:

\[
\sqrt{80x^4y^3} = \sqrt{80} \cdot \sqrt{x^4} \cdot \sqrt{y^3}
\]

### Step 2: Simplify each part

- \(\sqrt{80}\): The prime factorization of 80 is \(80 = 16 \times 5\), so:

\[
\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}
\]

- \(\sqrt{x^4}\): The square root of \(x^4\) is \(x^2\) because:

\[
\sqrt{x^4} = x^2
\]

- \(\sqrt{y^3}\): We can rewrite \(y^3\) as \(y^2 \times y\), so:

\[
\sqrt{y^3} = \sqrt{y^2 \times y} = \sqrt{y^2} \times \sqrt{y} = y \sqrt{y}
\]

### Step 3: Combine everything

Now, combining all the simplified parts:

\[
\sqrt{80x^4y^3} = 4\sqrt{5} \cdot x^2 \cdot y\sqrt{y} = 4x^2y\sqrt{5y}
\]

### Final Answer:
The simplified expression is:

\[
4x^2y\sqrt{5y}
\]

So, the correct answer is:

\[
\boxed{4x^2|y|\sqrt{5y}}
\]

This matches the first option in the multiple-choice list.

Would you like any further clarification on this problem? 

Here are five related questions to extend your understanding:
1. How would the simplification change if the expression was \( \sqrt{80x^6y^3} \)?
2. What is the square root of \(x^6y^5\)?
3. How do you simplify \( \sqrt{200x^8y^2} \)?
4. How would the expression simplify if \(y\) were negative?
5. What are the general steps to simplify square roots involving exponents?

**Tip**: When simplifying radicals, always check for perfect squares among the coefficients and variables to break them down efficiently.

Learn how to simplify the radical expression √80x^4y^3 with a detailed step-by-step solution. This problem covers key algebraic concepts such as simplifying square roots, working with exponents, and applying properties of square roots. Suitable for students in Grades 9-12.

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