Let's solve the given expression step-by-step to determine which option is equivalent to
[tex]\[
\left(\frac{y^{-3} z^5}{z^{-4} y^6}\right)^{-2}
\][/tex]
Step 1: Simplify the inner fraction
The given expression inside the parentheses is
[tex]\[
\frac{y^{-3} z^5}{z^{-4} y^6}
\][/tex]
We can simplify this by handling the base variables separately:
1. Simplify the terms with [tex]\( y \)[/tex]:
[tex]\[
\frac{y^{-3}}{y^6} = y^{-3-6} = y^{-9}
\][/tex]
2. Simplify the terms with [tex]\( z \)[/tex]:
[tex]\[
\frac{z^5}{z^{-4}} = z^{5-(-4)} = z^{5+4} = z^9
\][/tex]
So the simplified inner fraction becomes:
[tex]\[
\frac{y^{-3} z^5}{z^{-4} y^6} = y^{-9} z^9
\][/tex]
Step 2: Apply the outer exponent -2
Next, apply the exponent of -2 to the simplified expression [tex]\(y^{-9} z^9\)[/tex]:
[tex]\[
(y^{-9} z^9)^{-2} = y^{-9 \times -2} \cdot z^{9 \times -2} = y^{18} \cdot z^{-18}
\][/tex]
Simplifying this further, we have:
[tex]\[
y^{18} z^{-18} = \frac{y^{18}}{z^{18}}
\][/tex]
Conclusion:
The expression [tex]\(\left(\frac{y^{-3} z^5}{z^{-4} y^6}\right)^{-2}\)[/tex] simplifies to [tex]\(\frac{y^{18}}{z^{18}}\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{\frac{y^{18}}{z^{18}}}
\][/tex]
