To rewrite the expression \( \sqrt[3]{81} \) as a product of an integer and another cube root, we can first factor \( 81 \).
Since \( 81 = 27 \times 3 \) and \( 27 \) is a perfect cube (\( 27 = 3^3 \)), we can proceed as follows:
\[ \sqrt[3]{81} = \sqrt[3]{27 \times 3} \]
Using the property of cube roots that states \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \), we can rewrite this as:
\[ \sqrt[3]{81} = \sqrt[3]{27} \times \sqrt[3]{3} \]
Since \( \sqrt[3]{27} = 3 \), we substitute this back in:
\[ \sqrt[3]{81} = 3 \times \sqrt[3]{3} \]
Therefore, \( \sqrt[3]{81} \) rewritten as a product of an integer and another irrational cube root is:
\[ 3 \sqrt[3]{3} \]