To rewrite \( \sqrt[3]{48} \) as a product of an integer and another irrational cube root, we can first factor 48 into \( 24 \times 2 \):
\[ \sqrt[3]{48} = \sqrt[3]{24 \times 2} = \sqrt[3]{24} \cdot \sqrt[3]{2} \]
Next, we can factor 24 further into \( 8 \times 3 \):
\[ \sqrt[3]{24} = \sqrt[3]{8 \times 3} = \sqrt[3]{8} \cdot \sqrt[3]{3} \]
Since \( \sqrt[3]{8} = 2 \), we have:
\[ \sqrt[3]{24} = 2 \cdot \sqrt[3]{3} \]
Putting it all together:
\[ \sqrt[3]{48} = 2 \cdot \sqrt[3]{24} = 2 \cdot 2 \cdot \sqrt[3]{3} \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{6} \]
Thus, we can express \( \sqrt[3]{48} \) as:
\[ 4 \sqrt[3]{6} \]
Among the given response options, the correct answer is:
412−−√3 \ (4 \sqrt[3]{12}).