To simplify the expression $\left[(x^{-1})^{-4}\right]^{-3}$, we need to follow the properties of exponents step by step.

Here is the step-by-step solution:

1. Simplify the inner exponentiation: $(x^{-1})^{-4}$

   Recall the property of exponents $(a^m)^n = a^{mn}$. Therefore: $(x^{-1})^{-4} = x^{-1 \cdot -4} = x^4$

2. Simplify the outer exponentiation: $\left[x^4\right]^{-3}$

   Again, using the property of exponents $(a^m)^n = a^{mn}$: $\left[x^4\right]^{-3} = x^{4 \cdot -3} = x^{-12}$

So, the simplified form of the given expression $\left[(x^{-1})^{-4}\right]^{-3}$ is: $x^{-12}$

Would you like any further details or have any questions about this solution?

Here are 8 related questions to expand on this information:

1. What is the general rule for simplifying expressions with negative exponents?
2. How would you simplify $(a^{-2})^{-5}$?
3. What does $x^{-12}$ represent in terms of a fraction?
4. How would the expression change if we had a different exponent, such as $\left[(x^{-1})^{2}\right]^{3}$?
5. Can you provide an example with a different base, like $\left[(y^{-1})^{-3}\right]^{-2}$?
6. How do you handle exponents when dealing with multiplication inside the brackets?
7. What happens if the base is a fraction instead of a variable?
8. How can we verify the final result using properties of exponents?

Tip: When dealing with nested exponents, always simplify from the innermost part outward, applying the properties of exponents step by step to avoid errors.