To rewrite the cube root of 189 as the product of an integer and another cube root, we will first factor 189 into its prime factors and then simplify the cube root.
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Factor 189: \[ 189 = 3^3 \times 7 \] We find that \(189\) can be expressed as the product of \(3^3\) and \(7\).
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Now we can express the cube root: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
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We can separate the cube root into two parts: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]
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Simplifying \(\sqrt[3]{3^3}\): \[ \sqrt[3]{3^3} = 3 \]
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Therefore, we can write: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
Thus, the final expression is: \[ \sqrt[3]{189} = 3 \cdot \sqrt[3]{7} \]