Select the correct answer.

Which expression is equivalent to [tex]$32 \sqrt[3]{18 y} \div 8 \sqrt[3]{3 y}$[/tex], if [tex]$y \neq 0$[/tex]?

A. [tex]$12 \sqrt[3]{2 y^2}$[/tex]

B. [tex][tex]$4 \sqrt[3]{6}$[/tex][/tex]

C. [tex]$4 \sqrt[3]{15 y}$[/tex]

D. [tex]$4 \sqrt[2]{6 y}$[/tex]



Answer :

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]