To find the simplest equivalent form of the given expression:
[tex]\[
\sqrt[3]{\frac{8 x^6 y^9}{27 y^3 x^3}}
\][/tex]
we'll perform the following steps:
1. Simplify the expression inside the cube root:
[tex]\[
\frac{8 x^6 y^9}{27 y^3 x^3}
\][/tex]
2. Separate the coefficients and the variables:
[tex]\[
\frac{8}{27} \cdot \frac{x^6}{x^3} \cdot \frac{y^9}{y^3}
\][/tex]
3. Simplify each part:
- For the coefficients:
[tex]\[
\frac{8}{27}
\][/tex]
- For the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{x^6}{x^3} = x^{6-3} = x^3
\][/tex]
- For the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y^9}{y^3} = y^{9-3} = y^6
\][/tex]
4. Combine the simplified parts:
[tex]\[
\frac{8}{27} \cdot x^3 \cdot y^6
\][/tex]
5. Apply the cube root to each part:
[tex]\[
\sqrt[3]{\frac{8}{27}} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{y^6}
\][/tex]
6. Simplify the individual cube roots:
- For the coefficients:
[tex]\[
\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}
\][/tex]
- For the [tex]\(x^3\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
- For the [tex]\(y^6\)[/tex]:
[tex]\[
\sqrt[3]{y^6} = y^{6/3} = y^2
\][/tex]
7. Combine the results of the cube roots:
[tex]\[
\frac{2}{3} \cdot x \cdot y^2 = \frac{2 x y^2}{3}
\][/tex]
Thus, the simplest equivalent form of the given expression is:
[tex]\[
\frac{2 x y^2}{3}
\][/tex]
Now, compare this with the options provided:
A. [tex]\(\frac{2 x^3 y}{3 x}\)[/tex]
B. [tex]\(\frac{2 x^2 y^2}{3 x}\)[/tex]
C. [tex]\(\frac{2 x^5 y^2}{3 y^3 z^5}\)[/tex]
D. [tex]\(\frac{8 x^3 y^2}{27 x^3}\)[/tex]
We see that option B is the closest match when you simplify inside:
[tex]\[
\frac{2 x y^2}{3}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{2 x^2 y^2}{3 x}}
\][/tex]
So, the final answer is:
[tex]\[
\boxed{B}
\][/tex]
