To rewrite the irrational cube root \( \sqrt[3]{189} \) as the product of an integer and another cube root, we can first factor 189.
The prime factorization of 189 is:
\[ 189 = 3^3 \times 7 \]
Now, we can separate the cube root:
\[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
This can be split into two parts:
\[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]
Since \( \sqrt[3]{3^3} = 3 \), we have:
\[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
Thus, \( \sqrt[3]{189} \) can be rewritten as:
\[ 3 \sqrt[3]{7} \]