The given mathematical expression is:

$\left( \frac{x^1 y^6 z^3}{(xy)^{-1} z^{-4}} \right)^{-1} = x^r y^s z^t$

We need to simplify the left-hand side and find the values of $r$, $s$, and $t$.

Step-by-step simplification:

1. Simplify the denominator: $(xy)^{-1} = x^{-1} y^{-1}$ Thus, the denominator becomes: $(xy)^{-1} z^{-4} = x^{-1} y^{-1} z^{-4}$

2. Rewrite the entire fraction: The expression inside the parentheses now looks like this: $\frac{x^1 y^6 z^3}{x^{-1} y^{-1} z^{-4}}$

3. Apply the laws of exponents: When dividing like bases, subtract the exponents: $\frac{x^1}{x^{-1}} = x^{1 - (-1)} = x^{2}$ $\frac{y^6}{y^{-1}} = y^{6 - (-1)} = y^{7}$ $\frac{z^3}{z^{-4}} = z^{3 - (-4)} = z^{7}$

   So the simplified fraction becomes: $x^2 y^7 z^7$

4. Apply the outer exponent (-1): Now, we apply the exponent $-1$ to each term: $(x^2 y^7 z^7)^{-1} = x^{-2} y^{-7} z^{-7}$

5. Final result: Comparing this with the right-hand side of the equation: $x^{-2} y^{-7} z^{-7} = x^r y^s z^t$

   We can now identify $r$, $s$, and $t$: $r = -2, \quad s = -7, \quad t = -7$

Thus, the values are:

• $r = -2$
• $s = -7$
• $t = -7$

Would you like more details or further clarifications on this solution?

Here are some related questions you might find helpful:

1. How do you apply the laws of exponents when multiplying or dividing terms?
2. What happens when you raise a fraction to a negative exponent?
3. How does simplifying expressions with multiple variables work?
4. What are common mistakes to avoid when simplifying exponent expressions?
5. How do you factor exponents when they appear inside parentheses?

Tip: Always double-check the signs when applying the laws of exponents, especially with negative exponents.