To express the irrational cube root \( \sqrt[3]{3} - 24 \) as the product of an integer and another irrational cube root, we'll manipulate the expression.
First, we can factor out \( -24 \):
\[ \sqrt[3]{3} - 24 = -24 + \sqrt[3]{3} \]
However, to express this in the desired form directly isn't possible because it's a combination of an irrational number and an integer rather than a product directly.
If you meant to represent \( \sqrt[3]{3} - 24\) in a way where we only use the cube root and write it as a product to isolate the cube root, we can rewrite it as:
\[ = -24 \left( 1 - \frac{\sqrt[3]{3}}{24} \right) \]
This isn't exactly in the form of an integer directly multiplied by both irrational cube roots, but it highlights the integer multiplier of your cube root.
If you meant to approach something more specific, please clarify, and I can provide more targeted help!