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?

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.