To determine which of the given expressions is equal to [tex]\(\left[\frac{\left(x^2 y^3\right)^{-2}}{\left(x^6 y^3 z\right)^2}\right]\)[/tex], we need to simplify the given expression step-by-step.
Given expression:
[tex]\[ \left[\frac{\left(x^2 y^3\right)^{-2}}{\left(x^6 y^3 z\right)^2}\right] \][/tex]
Let's simplify [tex]\(\left(x^2 y^3\right)^{-2}\)[/tex]:
[tex]\[
\left(x^2 y^3\right)^{-2} = \frac{1}{\left(x^2 y^3\right)^2} = \frac{1}{x^{2 \cdot 2} y^{3 \cdot 2}} = \frac{1}{x^4 y^6}
\][/tex]
Next, simplify [tex]\(\left(x^6 y^3 z\right)^2\)[/tex]:
[tex]\[
\left(x^6 y^3 z\right)^2 = \left(x^6\right)^2 \left(y^3\right)^2 \left(z\right)^2 = x^{6 \cdot 2} y^{3 \cdot 2} z^{1 \cdot 2} = x^{12} y^6 z^2
\][/tex]
Now, we need to divide [tex]\(\frac{1}{x^4 y^6}\)[/tex] by [tex]\(x^{12} y^6 z^2\)[/tex]:
[tex]\[
\frac{\frac{1}{x^4 y^6}}{x^{12} y^6 z^2} = \frac{1}{x^4 y^6} \cdot \frac{1}{x^{12} y^6 z^2} = \frac{1}{x^{4+12} y^{6+6} z^2} = \frac{1}{x^{16} y^{12} z^2}
\][/tex]
Therefore, we have:
[tex]\[ \left[\frac{\left(x^2 y^3\right)^{-2}}{\left(x^6 y^3 z\right)^2}\right] = \frac{1}{x^{16} y^{12} z^2}\][/tex]
Comparing this result with the given options, we find that the correct option is:
[tex]\[ \boxed{\frac{1}{x^{16} y^{12} z^2}} \][/tex]
