To determine the expression equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex], we will simplify the expression step by step.
1. Simplify the base expression [tex]\(\frac{x^{-4} y}{x^{-9} y^5}\)[/tex]:
- The numerator is [tex]\(x^{-4} y\)[/tex].
- The denominator is [tex]\(x^{-9} y^5\)[/tex].
2. Combine the like bases in the fraction:
- For the [tex]\(x\)[/tex] terms, we use the rule of exponents [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex]:
[tex]\[
\frac{x^{-4}}{x^{-9}} = x^{-4 - (-9)} = x^{-4 + 9} = x^5
\][/tex]
- For the [tex]\(y\)[/tex] terms, we use the same exponent rule:
[tex]\[
\frac{y}{y^5} = y^{1-5} = y^{-4}
\][/tex]
3. Combine the results:
[tex]\[
\frac{x^{-4} y}{x^{-9} y^5} = x^5 \cdot y^{-4}
\][/tex]
4. Apply the outer exponent [tex]\(-2\)[/tex] to the simplified base:
- For [tex]\(x\)[/tex], we use the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(x^5)^{-2} = x^{5 \cdot -2} = x^{-10}
\][/tex]
- For [tex]\(y\)[/tex], we use the same rule:
[tex]\[
(y^{-4})^{-2} = y^{-4 \cdot -2} = y^{8}
\][/tex]
5. Combine the results of the power application:
[tex]\[
(x^5 \cdot y^{-4})^{-2} = x^{-10} \cdot y^8
\][/tex]
6. Express the result as a fraction:
[tex]\[
x^{-10} \cdot y^8 = \frac{y^8}{x^{10}}
\][/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].
Thus, the correct answer is [tex]\(\boxed{\frac{y^8}{x^{10}}}\)[/tex].
