https://file.aiquickdraw.com/mathbot/file/1728180771031f3wfuz15.jpg

Let's analyze the expression in the image:

\[
\left( \frac{x^{-3} y^3 z^{-4}}{y^{-3} z^4 x^{-5}} \right)^{-1}
\]

We need to simplify this step by step:

### Step 1: Simplify the fraction inside the parentheses

In the numerator, we have \( x^{-3}, y^3, z^{-4} \).  
In the denominator, we have \( y^{-3}, z^4, x^{-5} \).

Using the rule of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we simplify each base:

- For \(x\): \( \frac{x^{-3}}{x^{-5}} = x^{-3 - (-5)} = x^{2} \)
- For \(y\): \( \frac{y^3}{y^{-3}} = y^{3 - (-3)} = y^{6} \)
- For \(z\): \( \frac{z^{-4}}{z^{4}} = z^{-4 - 4} = z^{-8} \)

This simplifies the expression inside the parentheses to:

\[
x^2 y^6 z^{-8}
\]

### Step 2: Apply the outer exponent \(-1\)

Now, we need to apply the exponent \(-1\) to the whole expression. This will invert the signs of all the exponents:

\[
(x^2 y^6 z^{-8})^{-1} = x^{-2} y^{-6} z^{8}
\]

### Step 3: Match the given format

The expression equals \( \frac{z^r}{x^s y^t} \), so we need to write it in that format:

\[
\frac{z^{8}}{x^{2} y^{6}}
\]

From this, we can see that:
- \( r = 8 \) (the exponent of \(z\)),
- \( s = 2 \) (the exponent of \(x\)),
- \( t = 6 \) (the exponent of \(y\)).

### Conclusion:

- The exponent of \( z \) (\(r\)) is \( 8 \),
- The exponent of \( x \) (\(s\)) is \( 2 \) (which is correctly marked),
- The exponent of \( y \) (\(t\)) is \( 6 \).

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How do you simplify expressions involving negative exponents?
2. What are the rules for multiplying and dividing exponents with the same base?
3. How does raising a product to a power affect the exponents inside the product?
4. What happens when you raise a fraction to a negative exponent?
5. How can simplifying expressions with exponents help in solving equations?

**Tip**: Always simplify the expression inside parentheses first before applying outer exponents to avoid confusion.

The expression ((x^(-3) y^3 z^(-4)) / (y^(-3) z^4 x^(-5)))^(-1). Find the exponents of x, y, and z.

Learn how to simplify a rational expression involving negative exponents step-by-step. This problem covers key algebraic concepts, including properties of exponents and simplification techniques. Suitable for students in Grades 8-10.

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