The expression in the image is:

$\left\{ \left(\frac{1}{3}\right)^{-1} - \left(\frac{1}{4}\right)^{-1} \right\}^{-1}$

Let's break it down step by step:

Step 1: Simplify the individual terms inside the curly brackets.

1. Simplify $\left(\frac{1}{3}\right)^{-1}$: $\left(\frac{1}{3}\right)^{-1} = 3$ This is because raising a fraction to the power of $-1$ inverts the fraction.

2. Simplify $\left(\frac{1}{4}\right)^{-1}$: $\left(\frac{1}{4}\right)^{-1} = 4$

Step 2: Perform the subtraction inside the curly brackets.

$3 - 4 = -1$

Step 3: Apply the inverse operation to the result.

$(-1)^{-1} = -1$

Thus, the simplified value of the expression is $-1$.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How do you calculate the inverse of a fraction?
2. What is the general rule for negative exponents?
3. How does the subtraction of fractions differ from subtracting whole numbers?
4. What happens when you raise a negative number to an odd exponent?
5. Can this method be applied to more complex expressions involving multiple terms?

Tip: Always simplify each component of an expression step by step to avoid errors, especially when dealing with exponents and inverse operations.