Let's analyze the expression [tex]\(\left[\frac{1}{(x^6 y^3 z)^2}\right]\)[/tex] and simplify it step-by-step.
First, simplify the base inside the parentheses and then apply the exponentiation:
[tex]\[
(x^6 y^3 z)^2 = (x^6)^2 (y^3)^2 (z)^2 = x^{12} y^6 z^2
\][/tex]
So our expression becomes:
[tex]\[
\frac{1}{x^{12} y^6 z^2}
\][/tex]
Now, let's match this simplified expression with the provided options one by one.
### Option 1:
[tex]\[
\frac{(x^2 y^3)^{-6}}{(x^6 y^3 z)^6}
\][/tex]
First, calculate [tex]\((x^2 y^3)^{-6}\)[/tex]:
[tex]\[
(x^2 y^3)^{-6} = \frac{1}{(x^2 y^3)^6} = \frac{1}{x^{12} y^{18}}
\][/tex]
Then compute [tex]\((x^6 y^3 z)^6\)[/tex]:
[tex]\[
(x^6 y^3 z)^6 = (x^6)^6 (y^3)^6 (z)^6 = x^{36} y^{18} z^6
\][/tex]
Putting these together:
[tex]\[
\frac{\frac{1}{x^{12} y^{18}}}{x^{36} y^{18} z^6} = \frac{1}{x^{12} y^{18}} \cdot \frac{1}{x^{36} y^{18} z^6} = \frac{1}{x^{12+36} y^{18+18} z^6} = \frac{1}{x^{48} y^{36} z^6}
\][/tex]
This simplifies to:
[tex]\[
\frac{1}{x^{48} y^{36} z^6}
\][/tex]
which matches the simplified expression [tex]\(\frac{1}{x^{12} y^6 z^2}\)[/tex] exactly.
### Option 2:
[tex]\[
\frac{1}{x^{48} y^{36} z^6}
\][/tex]
This matches exactly what we found from the option 1 analysis above as the simplified expression is [tex]\(\frac{1}{x^{48} y^{36} z^6}\)[/tex]. Hence, this is a match.
### Option 3:
[tex]\[
\frac{(x^2 y^3)}{(x^6 y^3 z)^5}
\][/tex]
First, expand the denominator [tex]\((x^6 y^3 z)^5\)[/tex]:
[tex]\[
(x^6 y^3 z)^5 = (x^6)^5 (y^3)^5 (z)^5 = x^{30} y^{15} z^5
\][/tex]
So, the expression becomes:
[tex]\[
\frac{x^2 y^3}{x^{30} y^{15} z^5} = x^{2-30} y^{3-15} z^{-5} = x^{-28} y^{-12} z^{-5} = \frac{1}{x^{28} y^{12} z^5}
\][/tex]
This does not match [tex]\(\frac{1}{x^{12} y^6 z^2}\)[/tex].
### Option 4:
[tex]\[
\frac{x^{-12} y^{-18}}{x^{36} y^{18} z^6}
\][/tex]
Simplify this:
[tex]\[
\frac{x^{-12} y^{-18}}{x^{36} y^{18} z^6} = x^{-12-36} y^{-18-18} z^{-6} = x^{-48} y^{-36} z^{-6} = \frac{1}{x^{48} y^{36} z^6}
\][/tex]
This matches the simplified expression [tex]\(\frac{1}{x^{48} y^{36} z^6}\)[/tex].
### Conclusion:
Options 2 and 4 are both simplified to [tex]\(\frac{1}{x^{48} y^{36} z^6}\)[/tex] which matches our target expression after simplification.
The correct answer is:
[tex]\[
\boxed{2} \text{ or } 4
\][/tex]
