https://file.aiquickdraw.com/mathbot/file/17306839025844qdr9lp8.jpg

The expression in the image is:

\[
\sqrt[3]{24x^5 y^3}
\]

To simplify this, we’ll break down the expression inside the cube root and simplify any perfect cubes.

### Step-by-Step Solution

1. **Factorize the Constants and Variables**:
   - \(24\) can be factored as \(8 \times 3\), where \(8\) is a perfect cube (since \(8 = 2^3\)).
   - We have \(x^5\) and \(y^3\), where \(y^3\) is already a perfect cube.

   So we can rewrite the expression as:
   \[
   \sqrt[3]{8 \cdot 3 \cdot x^5 \cdot y^3}
   \]

2. **Separate Perfect Cubes**:
   - We can split the expression into two parts, one containing perfect cubes and the other not:
   \[
   \sqrt[3]{8 \cdot y^3} \cdot \sqrt[3]{3 \cdot x^5}
   \]

3. **Simplify the Cube Roots**:
   - For the perfect cubes, we have:
     \[
     \sqrt[3]{8 \cdot y^3} = 2y
     \]
   - For the remaining part \(\sqrt[3]{3 \cdot x^5}\), we can simplify \(x^5\) by separating it as \(x^3 \cdot x^2\):
     \[
     \sqrt[3]{3 \cdot x^3 \cdot x^2} = x \cdot \sqrt[3]{3x^2}
     \]

4. **Combine the Results**:
   - Now, combining everything, we get:
     \[
     2y \cdot x \cdot \sqrt[3]{3x^2} = 2xy \sqrt[3]{3x^2}
     \]

### Final Answer
\[
2xy \sqrt[3]{3x^2}
\]

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

### Related Questions
1. How would you simplify \(\sqrt[3]{54x^6 y^3}\)?
2. What is the process for simplifying \(\sqrt[3]{16x^3y^6}\)?
3. Can we simplify \(\sqrt[3]{81x^9 y^2}\) further?
4. What steps are involved in simplifying \(\sqrt[3]{48x^7 y^3 z^6}\)?
5. How would the simplification change if it were a fourth root instead of a cube root?

### Tip
When simplifying roots, it helps to factor out perfect powers (e.g., perfect squares, cubes) to simplify the expression.

Explore how to simplify the cube root expression \(\sqrt[3]{24x^5 y^3}\) with a clear, step-by-step solution. This problem involves key concepts in algebra, including properties of roots and radicals. Ideal for high school students in Grades 10-12.

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