To rewrite the irrational cube root \( \sqrt[3]{189} \) as the product of an integer and another cube root, we can start by factoring the number under the cube root.
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Factor 189: \[ 189 = 3^3 \times 7 \]
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Now, we can express \( \sqrt[3]{189} \) using this factorization: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
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Using the property of cube roots \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \), we can separate the terms: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]
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\( \sqrt[3]{3^3} \) simplifies to 3: \[ \sqrt[3]{3^3} = 3 \]
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Therefore, we can express \( \sqrt[3]{189} \) as: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
In conclusion, \( \sqrt[3]{189} \) can be rewritten as the product of an integer and another cube root: \[ \sqrt[3]{189} = 3\sqrt[3]{7} \]