To solve the question, we are tasked with rewriting the cube root of $48$ as the product of an integer and another irrational cube root.

Step-by-Step Solution:

1. Break down $48$ into a product of a perfect cube and another factor:

   • $48 = 8 \times 6$, where $8$ is a perfect cube ($2^3$).

2. Apply the cube root property:

   • $\sqrt[3]{48} = \sqrt[3]{8 \times 6} = \sqrt[3]{8} \cdot \sqrt[3]{6}$.

3. Simplify $\sqrt[3]{8}$:

   • $\sqrt[3]{8} = 2$.

4. Rewrite the expression:

   • $\sqrt[3]{48} = 2 \cdot \sqrt[3]{6}$.

Final Answer:

The correct choice is: $\boxed{2\sqrt[3]{6}}$

Let me know if you have questions or would like further clarification!

Follow-Up Questions:

1. Why is it important to identify a perfect cube in the factorization process?
2. Can this method be applied to square roots? Why or