To rewrite the cube root of \(189\) as the product of an integer and another cube root, we need to factor \(189\) in a way that includes a perfect cube.
First, let's factor \(189\):
\[ 189 = 3 \times 63 \] \[ 63 = 3 \times 21 \] \[ 21 = 3 \times 7 \]
So, we can write:
\[ 189 = 3^3 \times 7 \]
Now we can use this factorization to rewrite \(\sqrt[3]{189}\):
\[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
Using the property of cube roots that states \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\), we can separate the perfect cube from the rest:
\[ \sqrt[3]{189} = \sqrt[3]{3^3} \cdot \sqrt[3]{7} \]
Since \(\sqrt[3]{3^3} = 3\), we have:
\[ \sqrt[3]{189} = 3 \cdot \sqrt[3]{7} \]
Thus, the final product is:
\[ \sqrt[3]{189} = 3 \cdot \sqrt[3]{7} \]
This shows that \( \sqrt[3]{189} \) can be expressed as the product of the integer \(3\) and the irrational cube root \( \sqrt[3]{7} \).
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