To determine which expression is equivalent to [tex]\( 32 \sqrt[3]{18 y} \div 8 \sqrt[3]{3 y} \)[/tex], let's break down the given expression step-by-step:
1. Simplify the Coefficient:
We start by simplifying the numerical coefficients:
[tex]\[
\frac{32}{8} = 4
\][/tex]
2. Simplify the Cubic Roots:
For the cubic roots, we need to simplify:
[tex]\[
\frac{\sqrt[3]{18 y}}{\sqrt[3]{3 y}}
\][/tex]
Using the property of radicals, we can write this as a single cubic root:
[tex]\[
\sqrt[3]{\frac{18 y}{3 y}}
\][/tex]
Inside the cubic root, the [tex]\( y \)[/tex] terms cancel out, leaving:
[tex]\[
\sqrt[3]{\frac{18}{3}} = \sqrt[3]{6}
\][/tex]
3. Combine the Simplified Components:
Now we combine the simplified coefficient with the simplified cubic root:
[tex]\[
4 \cdot \sqrt[3]{6 y}
\][/tex]
Thus, the expression [tex]\( 32 \sqrt[3]{18 y} \div 8 \sqrt[3]{3 y} \)[/tex] is equivalent to [tex]\( 4 \sqrt[3]{6 y} \)[/tex].
Therefore, the correct choice is:
[tex]\[
\boxed{4 \sqrt[3]{6 y}}
\][/tex]
This matches option D.
