Let's simplify the given expression step-by-step to identify the correct answer:
[tex]\[ 32 \sqrt[3]{18 y} \div 8 \sqrt[3]{3 y} \][/tex]
### Step 1: Simplify the Fraction Outside the Cube Root
First, simplify the fraction [tex]\(\frac{32}{8}\)[/tex]:
[tex]\[ \frac{32}{8} = 4 \][/tex]
So, the expression becomes:
[tex]\[ 4 \left( \frac{\sqrt[3]{18 y}}{\sqrt[3]{3 y}} \right) \][/tex]
### Step 2: Simplify Inside the Cube Root
Next, we can combine the terms inside the cube roots:
[tex]\[ \frac{\sqrt[3]{18 y}}{\sqrt[3]{3 y}} = \sqrt[3]{\frac{18 y}{3 y}} \][/tex]
Since [tex]\( y \neq 0 \)[/tex], we can cancel [tex]\( y \)[/tex] in the numerator and denominator:
[tex]\[ \sqrt[3]{\frac{18 y}{3 y}} = \sqrt[3]{\frac{18}{3}} \][/tex]
Simplify the fraction [tex]\(\frac{18}{3}\)[/tex]:
[tex]\[ \frac{18}{3} = 6 \][/tex]
So, our expression now is:
[tex]\[ 4 \sqrt[3]{6} \][/tex]
### Conclusion
Comparing this result to the given choices:
A. [tex]\( 12 \sqrt[3]{2 y^2} \)[/tex]
B. [tex]\( 4 \sqrt[3]{6} \)[/tex]
C. [tex]\( 4 \sqrt[3]{15 y} \)[/tex]
D. [tex]\( 4 \sqrt[3]{6 y} \)[/tex]
The correct expression is [tex]\( 4 \sqrt[3]{6} \)[/tex], which matches Answer choice B.
