To rewrite the irrational cube root \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we can start by factoring \(189\) into its prime factors.
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Find the prime factorization of \(189\): \[ 189 = 3 \times 63 \] Continuing to factor \(63\): \[ 63 = 3 \times 21 \] And factoring \(21\): \[ 21 = 3 \times 7 \] Therefore, combining all these factors, we get: \[ 189 = 3^3 \times 7 \]
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Rewrite \(\sqrt[3]{189}\) using its prime factorization: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
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Use the property of cube roots: Using the property \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\), we can split the cube root: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]
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Simplify \(\sqrt[3]{3^3}\): Since \(\sqrt[3]{3^3} = 3\), we can write: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
So, the final result is: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
This shows \(\sqrt[3]{189}\) as the product of an integer \(3\) and another irrational cube root \(\sqrt[3]{7}\).