To rewrite the expression \( \sqrt[3]{189} \) as the product of an integer and another irrational cube root, we need to factor 189 into its prime factors and find perfect cube factors.
-
Finding the prime factorization of 189: \[ 189 \div 3 = 63 \ 63 \div 3 = 21 \ 21 \div 3 = 7 \] So, the prime factorization of 189 is: \[ 189 = 3^3 \times 7 \]
-
Identifying the perfect cube factor: The perfect cube in the factorization is \( 3^3 \).
-
Rewriting the cube root: We can rewrite \( \sqrt[3]{189} \) as: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \] Using the property of cube roots, this can be separated into: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]
-
Calculating the cube root of the perfect cube: \[ \sqrt[3]{3^3} = 3 \]
-
Combining the results: Therefore, we have: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]
Thus, the expression \( \sqrt[3]{189} \) can be rewritten as the product of an integer and another irrational cube root: \[ \sqrt[3]{189} = 3 \sqrt[3]{7} \]