To express the irrational cube root \( \sqrt[3]{189} \) as the product of an integer and another irrational cube root, we first need to factor 189 into its prime factors. The prime factorization of 189 is \( 3^3 \times 7 \). We can extract the cube root of \( 3^3 \), which is \( 3 \), leaving us with the cube root of \( 7 \) remaining inside the radical. Therefore, we can rewrite \( \sqrt[3]{189} \) as follows:
\[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} = \sqrt[3]{3^3} \times \sqrt[3]{7} = 3 \times \sqrt[3]{7}. \]
Thus, \( \sqrt[3]{189} \) can be expressed as \( 3\sqrt[3]{7} \). This decomposition highlights how we can separate the rational factors from the irrational factors while still preserving the equality of the expression. The integer \( 3 \) is the cube root of the perfect cube \( 27 \), and \( \sqrt[3]{7} \) remains irrational since \( 7 \) is not a perfect cube. Hence, our final expression \( 3\sqrt[3]{7} \) elegantly combines an integer and an irrational cube root, clearly illustrating the intended form.