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The given expression for \( A \) is:

\[
A = \left( \frac{x^2}{y^3} \right)^{-5} \left( \frac{x^{-3}}{y^4} \right)^3 \cdot \left( x^3 y^{-1} \right)
\]

Let's simplify this step-by-step:

### 1. Simplifying Each Term
1. **First Term:** 
   \[
   \left( \frac{x^2}{y^3} \right)^{-5} = \frac{y^{15}}{x^{10}}
   \]
   (by applying the negative exponent rule, \( (a/b)^{-n} = (b/a)^n \)).

2. **Second Term:** 
   \[
   \left( \frac{x^{-3}}{y^4} \right)^3 = \frac{x^{-9}}{y^{12}}
   \]
   (by raising both numerator and denominator to the power of 3).

3. **Third Term:** 
   \[
   x^3 y^{-1} = \frac{x^3}{y}
   \]

### 2. Combining the Terms
Now, let's combine all three simplified terms:

\[
A = \frac{y^{15}}{x^{10}} \cdot \frac{x^{-9}}{y^{12}} \cdot \frac{x^3}{y}
\]

Simplifying step-by-step:

1. Combine all \( x \)-terms:
   \[
   \frac{1}{x^{10}} \cdot \frac{1}{x^9} \cdot x^3 = \frac{x^{-16} \cdot x^3}{1} = \frac{x^{-13}}{1} = \frac{1}{x^{13}}
   \]

2. Combine all \( y \)-terms:
   \[
   \frac{y^{15}}{1} \cdot \frac{1}{y^{12}} \cdot \frac{1}{y} = \frac{y^{15}}{y^{13}} = y^2
   \]

### 3. Final Simplified Form
Combining these results, we have:

\[
A = \frac{y^2}{x^{13}}
\]

This is the simplified form of the given expression.

---

Do you need more details or have any questions?

Here are some related questions:

1. How do you handle negative exponents when simplifying expressions?
2. What are the steps to simplify a complex fraction?
3. How do you multiply and divide powers with the same base?
4. How does the distribution of exponents work over fractions?
5. Can this type of simplification be applied to complex numbers?

**Tip:** Always apply exponent rules carefully when combining terms, especially with negative exponents, to avoid errors in simplification.

Simplify the expression A = ((x^2/y^3)^(-5)) * ((x^(-3)/y^4)^3) * (x^3 * y^(-1)).

Learn how to simplify the algebraic expression A = ((x^2/y^3)^(-5)) * ((x^(-3)/y^4)^3) * (x^3 * y^(-1)) using exponent rules. This detailed solution walks through simplifying complex algebraic expressions, focusing on key exponent rules and algebraic simplification techniques. Suitable for Grades 9-12.

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