If `((a^(-1)b^(2))/(a^(2)b^(-4)))div((a^(3)b^(-5))/(a^(-2)b^(3)))=a^(x).b^(y),` find x + y.

Answer

To solve the equation

\[ \frac{\frac{a^{-1}b^{2}}{a^{2}b^{-4}}}{\frac{a^{3}b^{-5}}{a^{-2}b^{3}}} = a^{x}b^{y} \]

and find \(x + y\), we will follow these steps:

### Step 1: Simplify the numerator and denominator

Start with the expression:

\[ \frac{a^{-1}b^{2}}{a^{2}b^{-4}} \]

This can be simplified by applying the property of exponents:

\[ = a^{-1}b^{2} \cdot a^{-2}b^{4} = a^{-1 - 2}b^{2 + 4} = a^{-3}b^{6} \]

Now, simplify the denominator:

\[ \frac{a^{3}b^{-5}}{a^{-2}b^{3}} = a^{3}b^{-5} \cdot a^{2}b^{-3} = a^{3 + 2}b^{-5 - 3} = a^{5}b^{-8} \]

### Step 2: Rewrite the entire expression

Now, we can rewrite the entire expression:

\[ \frac{a^{-3}b^{6}}{a^{5}b^{-8}} = a^{-3}b^{6} \cdot a^{-5}b^{8} = a^{-3 - 5}b^{6 + 8} = a^{-8}b^{14} \]

### Step 3: Set the expression equal to \(a^{x}b^{y}\)

Now we have:

\[ a^{-8}b^{14} = a^{x}b^{y} \]

### Step 4: Compare the exponents

From the equation above, we can compare the exponents:

- For \(a\): \(x = -8\)
- For \(b\): \(y = 14\)

### Step 5: Calculate \(x + y\)

Now, we need to find \(x + y\):

\[ x + y = -8 + 14 = 6 \]

Thus, the final answer is:

\[ \boxed{6} \]