To rewrite the expression \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we start by factoring \(189\) into its prime factors.
First, we can find the prime factorization of \(189\):
\[ 189 = 3 \times 63 = 3 \times 3 \times 21 = 3^3 \times 7 \]
Thus, we can express \(189\) as:
\[ 189 = 3^3 \times 7 \]
Now, we can rewrite \(\sqrt[3]{189}\):
\[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
Using the property of cube roots that states \(\sqrt[3]{a \times b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\), we can split the expression:
\[ \sqrt[3]{189} = \sqrt[3]{3^3} \cdot \sqrt[3]{7} \]
Calculating \(\sqrt[3]{3^3}\), we find:
\[ \sqrt[3]{3^3} = 3 \]
Thus, the expression simplifies to:
\[ \sqrt[3]{189} = 3 \cdot \sqrt[3]{7} \]
So, we have rewritten \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root:
\[ \sqrt[3]{189} = 3\sqrt[3]{7} \]