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.