To simplify the given expression [tex]\(\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}\)[/tex], we will follow a series of steps working inside the cube roots and then simplifying.
First, notice that the expression inside the cube roots can be simplified by combining the fractions:
[tex]\[
\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = \sqrt[3]{\frac{32 x^3 y^6}{2 x^9 y^2}}
\][/tex]
Now, simplify the fraction inside the cube root:
1. Separate the numerical coefficients:
[tex]\[
\frac{32}{2} = 16
\][/tex]
2. Simplify the powers of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^3}{x^9} = x^{3-9} = x^{-6} = \frac{1}{x^6}
\][/tex]
3. Simplify the powers of [tex]\(y\)[/tex]:
[tex]\[
\frac{y^6}{y^2} = y^{6-2} = y^4
\][/tex]
Putting all these together, we get:
[tex]\[
\frac{32 x^3 y^6}{2 x^9 y^2} = 16 \cdot \frac{1}{x^6} \cdot y^4 = \frac{16 y^4}{x^6}
\][/tex]
So, the simplified expression inside the cube root is:
[tex]\[
\sqrt[3]{\frac{16 y^4}{x^6}}
\][/tex]
Therefore, the simplified form of the original expression is:
[tex]\[
\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = \sqrt[3]{\frac{16 y^4}{x^6}}
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\sqrt[3]{\frac{16 y^4}{x^6}}
\][/tex]
So, the correct answer is:
[tex]\(\boxed{\sqrt[3]{\frac{16 y^4}{x^6}}}\)[/tex]
