To rewrite the irrational cube root \( \sqrt[3]{189} \) as the product of an integer and another irrational cube root, we can start by simplifying \( 189 \).
First, we can factor \( 189 \) into its prime factors: \[ 189 = 3^3 \times 7. \]
Next, we can apply the property of cube roots to separate the factors: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7}. \]
Now, we can use the property of cube roots that states: \[ \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}. \] Using this property, we can rewrite \( \sqrt[3]{189} \): \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7}. \]
Since \( \sqrt[3]{3^3} = 3 \), we have: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7}. \]
Thus, we have rewritten \( \sqrt[3]{189} \) as the product of an integer and another irrational cube root: \[ \sqrt[3]{189} = 3\sqrt[3]{7}. \]