Alright, let's break down the expression step by step to find an equivalent expression for [tex]\(32 \sqrt[3]{18 y} \div 8 \sqrt[3]{3 y}\)[/tex].
1. Simplify the constants: Look at the constant multipliers separately.
[tex]\[
\frac{32}{8} = 4
\][/tex]
So the initial expression reduces to:
[tex]\[
4 \cdot \frac{\sqrt[3]{18 y}}{\sqrt[3]{3 y}}
\][/tex]
2. Simplify the cube roots:
Now, focus on the fraction within the cube roots:
[tex]\[
\frac{\sqrt[3]{18 y}}{\sqrt[3]{3 y}}
\][/tex]
We can use the property of cube roots that states [tex]\(\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}\)[/tex]. Applying this property, we have:
[tex]\[
\frac{\sqrt[3]{18 y}}{\sqrt[3]{3 y}} = \sqrt[3]{\frac{18 y}{3 y}}
\][/tex]
Simplify the fraction inside the cube root:
[tex]\[
\frac{18 y}{3 y} = 6
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
\sqrt[3]{6}
\][/tex]
3. Combine the simplified constant with the cube root:
We previously simplified the constants to get the factor 4. Now, multiplying 4 by the simplified cube root, we have:
[tex]\[
4 \sqrt[3]{6}
\][/tex]
Thus, the expression [tex]\(32 \sqrt[3]{18 y} \div 8 \sqrt[3]{3 y}\)[/tex] simplifies to [tex]\(4 \sqrt[3]{6}\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{B. \, 4 \sqrt[3]{6}} \][/tex]
