The mathematical expression in the image is:

$\left( \frac{s^{4t-6}}{6s^{-1}t} \right)^{-2}$

To simplify this expression, we can follow these steps:

Step 1: Distribute the negative exponent $-2$ to both the numerator and the denominator:

$\left( \frac{s^{4t-6}}{6s^{-1}t} \right)^{-2} = \frac{(s^{4t-6})^{-2}}{(6s^{-1}t)^{-2}}$

Step 2: Apply the power of a power rule $(a^m)^n = a^{m \cdot n}$:

$= \frac{s^{-(4t-6) \cdot 2}}{(6s^{-1}t)^{-2}}$ $= \frac{s^{-8t+12}}{(6s^{-1}t)^{-2}}$

Step 3: Simplify the denominator. For the product $6s^{-1}t$, apply the exponent to each factor:

$(6s^{-1}t)^{-2} = \frac{1}{6^2 \cdot s^{-2} \cdot t^2} = \frac{1}{36s^{-2}t^2}$

Step 4: Now rewrite the expression with the simplified denominator:

$= \frac{s^{-8t+12}}{\frac{1}{36s^{-2}t^2}}$

Step 5: To simplify the division, multiply by the reciprocal of the denominator:

$= s^{-8t+12} \cdot 36s^2t^2$ $= 36s^{-8t+12+2}t^2$

Step 6: Combine the exponents for $s$:

$= 36s^{-8t+14}t^2$

The final simplified expression is: $36s^{-8t+14}t^2$

Would you like more details on any step? Here are a few related questions:

1. What is the power of a power rule in exponents, and how does it apply?
2. How do you handle negative exponents in fractions?
3. Can you explain how to simplify products and powers in algebraic expressions?
4. What is the reciprocal rule in division of fractions?
5. How do exponents affect constants in algebraic terms?

Tip: When working with exponents, remember that a negative exponent represents the reciprocal of the base raised to the positive power.