To solve the given problem, we'll break it down into manageable steps and simplify the expressions. Let's start with the given expression:
[tex]\[
\left(\frac{a^{-1}b^2}{a^2b^{-4}}\right)^7 \div \left(\frac{a^3b^{-5}}{a^{-2}b^3}\right)^{-5}
\][/tex]
### Step 1: Simplify the Inner Expressions
First, simplify the numerator and denominator inside each fraction individually.
#### Expression 1: [tex]\(\frac{a^{-1}b^2}{a^2b^{-4}}\)[/tex]
Simplify the fraction:
[tex]\[
\frac{a^{-1}b^2}{a^2b^{-4}} = a^{-1 - 2} \cdot b^{2 - (-4)} = a^{-3} \cdot b^6
\][/tex]
#### Expression 2: [tex]\(\frac{a^3b^{-5}}{a^{-2}b^3}\)[/tex]
Simplify the fraction:
[tex]\[
\frac{a^3b^{-5}}{a^{-2}b^3} = a^{3 - (-2)} \cdot b^{-5 - 3} = a^5 \cdot b^{-8}
\][/tex]
### Step 2: Raise the Simplified Fractions to the Given Powers
Next, raise the simplified expressions to their respective exponents.
#### Raising Expression 1 to the 7th Power:
[tex]\[
(a^{-3}b^6)^7 = a^{-3 \cdot 7} \cdot b^{6 \cdot 7} = a^{-21} \cdot b^{42}
\][/tex]
#### Raising Expression 2 to the -5th Power:
[tex]\[
(a^5b^{-8})^{-5} = (a^5)^{-5} \cdot (b^{-8})^{-5} = a^{5 \cdot (-5)} \cdot b^{-8 \cdot (-5)} = a^{-25} \cdot b^{40}
\][/tex]
### Step 3: Combine the Expressions
Divide the two expressions:
[tex]\[
\frac{a^{-21}b^{42}}{a^{-25}b^{40}}
\][/tex]
This simplifies to:
[tex]\[
a^{-21 - (-25)} \cdot b^{42 - 40} = a^{-21 + 25} \cdot b^{42 - 40} = a^4 \cdot b^2
\][/tex]
### Step 4: Determine the Exponents and Sum
The expression simplifies to [tex]\(a^4 \cdot b^2\)[/tex]. Thus, the exponents are:
[tex]\[
x = 4 \quad \text{and} \quad y = 2
\][/tex]
Finally, calculate [tex]\(x + y\)[/tex]:
[tex]\[
x + y = 4 + 2 = 6
\][/tex]
### Conclusion
[tex]\[
\boxed{6}
\][/tex]
