To determine which expression is equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex], we will simplify the given expression step-by-step.
### Step 1: Simplify the Inner Fraction
Starting with the inner fraction:
[tex]\[
\frac{x^{-4} y}{x^{-9} y^5}
\][/tex]
We can handle each component separately:
#### 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]
Combining these simplified components, the inner fraction becomes:
[tex]\[
\frac{x^{-4} y}{x^{-9} y^5} = \frac{x^5}{y^4}
\][/tex]
### Step 2: Apply the Outer Exponent
Next, we need to apply the exponent [tex]\(-2\)[/tex] to the simplified inner fraction:
[tex]\[
\left(\frac{x^5}{y^4}\right)^{-2}
\][/tex]
Using the property of exponents [tex]\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)[/tex], we get:
[tex]\[
\left(\frac{x^5}{y^4}\right)^{-2} = \frac{(x^5)^{-2}}{(y^4)^{-2}}
\][/tex]
### Step 3: Simplify the Resulting Expression
Now, simplify the exponents:
[tex]\[
(x^5)^{-2} = x^{5 \cdot (-2)} = x^{-10}
\][/tex]
[tex]\[
(y^4)^{-2} = y^{4 \cdot (-2)} = y^{-8}
\][/tex]
So, we have:
[tex]\[
\frac{x^{-10}}{y^{-8}}
\][/tex]
### Step 4: Rewrite with Positive Exponents
To write this expression with positive exponents, we use the property [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]:
[tex]\[
\frac{x^{-10}}{y^{-8}} = x^{-10} \cdot y^8 = \frac{y^8}{x^{10}}
\][/tex]
Thus, the equivalent expression we were looking for is:
[tex]\[
\boxed{\frac{y^8}{x^{10}}}
\][/tex]
This matches the first option in the provided list, confirming that:
[tex]\[
\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2} \quad \text{is equivalent to} \quad \frac{y^8}{x^{10}}.
\][/tex]
