To simplify the given expression, we start with:
[tex]\[
\sqrt[3]{\frac{8 x^6 y^9}{27 y^3 z^3}}
\][/tex]
Step 1: Simplify the expression inside the cube root.
First, separate the constants and the variables:
For the constants:
[tex]\[
\frac{8}{27} = \frac{2^3}{3^3}
\][/tex]
For the variable with [tex]\(x\)[/tex]:
[tex]\[
x^6 \text{ remains as it is}
\][/tex]
For the variables with [tex]\(y\)[/tex]:
[tex]\[
\frac{y^9}{y^3} = y^{9-3} = y^6
\][/tex]
For the variable with [tex]\(z\)[/tex]:
[tex]\[
z^3 \text{ remains as it is}
\][/tex]
So now we can rewrite the expression inside the cube root as:
[tex]\[
\frac{2^3 x^6 y^6}{3^3 z^3}
\][/tex]
Step 2: Take the cube root of the simplified expression.
The cube root of a fraction [tex]\(\frac{a}{b}\)[/tex] can be written as [tex]\(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)[/tex].
Applying this property, we get:
[tex]\[
\sqrt[3]{\frac{2^3 x^6 y^6}{3^3 z^3}} = \frac{\sqrt[3]{2^3 x^6 y^6}}{\sqrt[3]{3^3 z^3}}
\][/tex]
Now, extract the cube roots:
[tex]\[
\sqrt[3]{2^3 x^6 y^6} = 2 x^2 y^2
\][/tex]
and
[tex]\[
\sqrt[3]{3^3 z^3} = 3 z
\][/tex]
So, the entire expression simplifies to:
[tex]\[
\frac{2 x^2 y^2}{3 z}
\][/tex]
Double check with the options given:
A. [tex]\(\frac{2 x^3 y}{3 x}\)[/tex]
B. [tex]\(\frac{2 x^2 y^2}{3 x}\)[/tex]
C. [tex]\(\frac{2 x^6 y^2}{3 y^3 x^3}\)[/tex]
D. [tex]\(\frac{8 x^3 y^2}{27 z^3}\)[/tex]
The simplified form matches none of the options precisely as given in the initial problem. However, correcting the form and verifying, we see:
The accurate and simplified answer matches:
[tex]\[
\frac{2 x^2 y^2}{3 z}
\][/tex]
Please verify this against the choices properly; there may have been an issue with options provided.
Based on accurate calculation and understanding of the cube roots:
The closest correct form should have been:
[tex]\[
B. \frac{2 x^2 y^2}{3 z}
\][/tex]
