To rewrite \( \sqrt[3]{81} \) as a product of an integer and another irrational cube root, we start by factoring \( 81 \).
We can express \( 81 \) as \( 27 \times 3 \), since \( 27 = 3^3 \) is a perfect cube.
Now, we can rewrite the cube root:
\[ \sqrt[3]{81} = \sqrt[3]{27 \times 3} \]
Using the property of cube roots, we can separate the product:
\[ \sqrt[3]{81} = \sqrt[3]{27} \times \sqrt[3]{3} \]
Since \( \sqrt[3]{27} = 3 \):
\[ \sqrt[3]{81} = 3 \times \sqrt[3]{3} \]
Therefore, \( \sqrt[3]{81} \) rewritten as a product of an integer and another irrational cube root is:
\[ 3 \sqrt[3]{3} \]