To solve the expression [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex], we will first simplify the fraction inside the parentheses and then apply the exponent [tex]\(-2\)[/tex] to the result.
1. Start with the expression inside the parentheses:
[tex]\[
\frac{x^{-4} y}{x^{-9} y^5}
\][/tex]
2. Simplify each part of the fraction:
- Simplify the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{x^{-4}}{x^{-9}} = x^{-4 - (-9)} = x^{-4 + 9} = x^5
\][/tex]
- Simplify the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y}{y^5} = y^{1-5} = y^{-4}
\][/tex]
3. Combine the simplified parts:
[tex]\[
\frac{x^{-4} y}{x^{-9} y^5} = x^5 y^{-4}
\][/tex]
4. Now, apply the exponent [tex]\(-2\)[/tex] to the simplified expression:
[tex]\[
\left(x^5 y^{-4}\right)^{-2}
\][/tex]
5. Distribute the exponent [tex]\(-2\)[/tex] to both [tex]\(x^5\)[/tex] and [tex]\(y^{-4}\)[/tex]:
- For [tex]\(x^5\)[/tex]:
[tex]\[
\left(x^5\right)^{-2} = x^{5 \cdot -2} = x^{-10}
\][/tex]
- For [tex]\(y^{-4}\)[/tex]:
[tex]\[
\left(y^{-4}\right)^{-2} = y^{-4 \cdot -2} = y^8
\][/tex]
6. Combine the results:
[tex]\[
x^{-10} y^8 = \frac{y^8}{x^{10}}
\][/tex]
Therefore, the expression equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex] is [tex]\(\frac{y^8}{x^{10}}\)[/tex].
The correct answer is:
[tex]\[
\boxed{\frac{y^8}{x^{10}}}
\][/tex]
