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The expression to simplify is:

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

First, simplify the base inside the parentheses before applying the exponent:

1. Combine the \(x\)'s: \( x^2 \cdot x^{-3} = x^{2 - 3} = x^{-1} \).
2. Combine the \(y\)'s: \( y^5 \cdot y^4 = y^{5 + 4} = y^9 \), and in the denominator \(y^4 \cdot y^4 = y^{8}\).
3. Combine the \(z\)'s: \( z^4 \) and \( z^3 \) in the denominator gives \( z^{4 - 3} = z^1 \).

So, the expression inside the parentheses becomes:

\[
\frac{x^{-1} y^9 z^1}{x^3 y^8}
\]

Simplify each variable:

\[
x^{-1 - 3} = x^{-4}, \quad y^{9 - 8} = y^1, \quad z^1 \text{ stays the same}.
\]

Now, we have:

\[
\frac{x^{-4} y^1 z^1}{1} = x^{-4} y z
\]

Now apply the \(-4\) exponent to the entire expression:

\[
(x^{-4} y z)^{-4} = x^{16} y^{-4} z^{-4}
\]

So, \(r = 16\), \(s = -4\), and \(t = -4\).

Would you like any further details or explanations?

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Here are 5 related questions for more practice:

1. How do you simplify a fraction with exponents in the numerator and denominator?
2. What are the rules for multiplying and dividing exponents?
3. How do negative exponents affect the base variable?
4. How can you apply the power rule to multiple variables in an expression?
5. What happens when you raise a power to another power?

**Tip:** When dealing with exponents, always combine like terms (same bases) before applying additional exponent rules!

The expression (x^2 y^5 z^4 x^-3) / (x^3 y^4 z^3 y^4)^(-4) equals x^r y^s z^t. Where r, the exponent of x, is: 
and s, the exponent of y, is: 
and finally t, the exponent of z, is:

Learn how to simplify the expression (x^2 y^5 z^4 x^-3) / (x^3 y^4 z^3 y^4)^(-4) using exponent rules. This problem covers key concepts such as the product and quotient of powers, and the power of a power. Suitable for students in Grades 8-10.

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