To rewrite the irrational cube root \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we first need to factor the number 189 into its prime factors.
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Factor 189: \[ 189 = 3 \times 63 \] Continuing to factor 63: \[ 63 = 3 \times 21 \] And continuing to factor 21: \[ 21 = 3 \times 7 \] Putting it all together, the prime factorization of 189 is: \[ 189 = 3^3 \times 7 \]
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Separate the factors: We can express \(\sqrt[3]{189}\) as follows: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]
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Use the property of cube roots: The property of cube roots states that \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\). Applying this property, we can separate the cube roots: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]
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Calculate the integer cube root: We know that \(\sqrt[3]{3^3} = 3\). Therefore, we have: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
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Final Expression: Thus, we can rewrite \(\sqrt[3]{189}\) as: \[ \sqrt[3]{189} = 3\sqrt[3]{7} \]
This shows that the irrational cube root \(\sqrt[3]{189}\) can be expressed as the product of an integer (3) and another irrational cube root (\(\sqrt[3]{7}\)).
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