To rewrite the irrational cube root \( \sqrt[3]{189} \) as the product of an integer and another irrational cube root, we'll start by factoring 189 into its prime factors.
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Factor 189:
- \( 189 = 3 \times 63 = 3 \times 3 \times 21 = 3^3 \times 7 \)
- Therefore, \( 189 = 3^3 \times 7 \).
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Now, we can rewrite \( \sqrt[3]{189} \): \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
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Apply the property of cube roots: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]
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Since \( \sqrt[3]{3^3} = 3 \), we can express it as: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
Thus, the expression \( \sqrt[3]{189} \) can be rewritten as the product of an integer and another irrational cube root: \[ \sqrt[3]{189} = 3 \cdot \sqrt[3]{7} \]