To simplify the expression
[tex]\[
\sqrt[3]{\frac{8 x^6 y^9}{27 y^3 z^3}},
\][/tex]
we need to follow a few algebraic steps carefully.
1. Simplify the Fraction Inside the Cube Root:
[tex]\[
\frac{8 x^6 y^9}{27 y^3 z^3}
\][/tex]
Notice that [tex]\( y^9 \)[/tex] and [tex]\( y^3 \)[/tex] can be simplified:
[tex]\[
\frac{8 x^6 y^9}{27 y^3 z^3} = \frac{8 x^6 \cdot y^9}{27 \cdot y^3 \cdot z^3} = \frac{8 x^6 \cdot y^{9-3}}{27 \cdot z^3} = \frac{8 x^6 \cdot y^6}{27 \cdot z^3}.
\][/tex]
2. Apply the Cube Root to Each Term:
[tex]\[
\sqrt[3]{\frac{8 x^6 y^6}{27 z^3}}
\][/tex]
This can be broken down into:
[tex]\[
\sqrt[3]{8} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^6} \cdot \sqrt[3]{\frac{1}{27 z^3}}
\][/tex]
3. Compute Each Cube Root:
- [tex]\(\sqrt[3]{8} = 2\)[/tex]
- [tex]\(\sqrt[3]{x^6} = x^{6/3} = x^2\)[/tex]
- [tex]\(\sqrt[3]{y^6} = y^{6/3} = y^2\)[/tex]
- [tex]\(\sqrt[3]{\frac{1}{27 z^3}} = \frac{1}{\sqrt[3]{27}} \cdot \frac{1}{\sqrt[3]{z^3}} = \frac{1}{3} \cdot \frac{1}{z} = \frac{1}{3z}\)[/tex]
4. Combine the Simplified Terms:
[tex]\[
2 \cdot x^2 \cdot y^2 \cdot \frac{1}{3z} = \frac{2 x^2 y^2}{3 z}
\][/tex]
Thus, the simplest equivalent form of the expression [tex]\(\sqrt[3]{\frac{8 x^6 y^9}{27 y^3 z^3}}\)[/tex] is:
[tex]\[
\boxed{\frac{2 x^2 y^2}{3 z}}
\][/tex]
So the correct answer is B. [tex]\(\frac{2 x^2 y^2}{3 z}\)[/tex].
