To rewrite \( \sqrt[3]{48} \) as a product of an integer and another irrational cube root, we start by factoring \( 48 \):
\[ 48 = 16 \times 3 = 4^2 \times 3 \]
Now we can express the cube root:
\[ \sqrt[3]{48} = \sqrt[3]{16 \times 3} = \sqrt[3]{16} \cdot \sqrt[3]{3} \]
Next, we can write \( 16 \) as \( 4^2 \):
\[ \sqrt[3]{16} = \sqrt[3]{4^2} = 4^{2/3} \]
However, since we need to express it as a simple integer coefficient times another cube root, we can take out the perfect cube from \( 48 \):
Notably, we can rewrite \( 48 \) as follows:
\[ 48 = 8 \times 6 \]
where \( 8 \) is a perfect cube (\( 2^3 \)):
\[ \sqrt[3]{48} = \sqrt[3]{8 \times 6} = \sqrt[3]{8} \cdot \sqrt[3]{6} = 2 \cdot \sqrt[3]{6} \]
Thus, we can express \( \sqrt[3]{48} \) as:
\[ \sqrt[3]{48} = 2 \sqrt[3]{6} \]
Therefore, the correct response is:
2 root index 3 start root 6 end root