((x^5 y^2 z^3 x^-2) / (x^3 y^5 z^4 y^4))^-4

Let's simplify the expression step by step:

### Expression:
\[
\left(\frac{x^5 y^2 z^3 x^{-2}}{x^3 y^5 z^4 y^4}\right)^{-4}
\]

### Step 1: Simplify the expression inside the parentheses
1. Combine like terms in the numerator:
   \[
   x^5 \cdot x^{-2} = x^{5-2} = x^3
   \]
   The numerator simplifies to \(x^3 y^2 z^3\).

2. Combine like terms in the denominator:
   \[
   y^5 \cdot y^4 = y^{5+4} = y^9
   \]
   The denominator is \(x^3 y^9 z^4\).

So, the expression inside the parentheses simplifies to:
\[
\frac{x^3 y^2 z^3}{x^3 y^9 z^4}
\]

### Step 2: Simplify further by canceling out common terms
1. Cancel out \(x^3\) in the numerator and denominator:
   \[
   \frac{x^3}{x^3} = 1
   \]
   This leaves:
   \[
   \frac{y^2 z^3}{y^9 z^4}
   \]

2. Simplify the remaining \(y\) and \(z\) terms:
   \[
   y^2 \div y^9 = y^{2-9} = y^{-7}
   \]
   \[
   z^3 \div z^4 = z^{3-4} = z^{-1}
   \]

Now, the expression inside the parentheses is:
\[
\frac{y^{-7} z^{-1}}{1} = y^{-7} z^{-1}
\]

### Step 3: Apply the exponent \(-4\)
\[
\left(y^{-7} z^{-1}\right)^{-4}
\]

Distribute the exponent \(-4\) across both \(y^{-7}\) and \(z^{-1}\):
\[
y^{-7 \times -4} z^{-1 \times -4} = y^{28} z^{4}
\]

### Final Answer:
\[
y^{28} z^{4}
\]

Would you like more details, or do you have any questions?

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**Related Questions:**
1. How would the expression change if the exponent outside the parentheses was positive instead of negative?
2. What would happen if there were additional variables in the numerator or denominator?
3. Can you simplify an expression like this if the bases (x, y, z) are different in the numerator and denominator?
4. How does the order of operations impact the simplification of such expressions?
5. What are the rules for dealing with negative exponents in more complex algebraic expressions?

**Tip:** Always simplify inside the parentheses first before applying any outside exponents.

Learn how to simplify the algebraic expression (x^5 y^2 z^3 x^-2) / (x^3 y^5 z^4 y^4)^-4 step-by-step. This problem involves exponent rules and simplification techniques. Suitable for high school students.

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