To solve the expression $\left(\frac{1}{4}\right)^{-1} - \left(\frac{1}{3}\right)^{-2}$, let's break it down step-by-step.

Step 1: Evaluate each term separately

1. First Term: $\left(\frac{1}{4}\right)^{-1}$

   • Using the property $a^{-1} = \frac{1}{a}$, we get: $\left(\frac{1}{4}\right)^{-1} = \frac{1}{\frac{1}{4}} = 4$

2. Second Term: $\left(\frac{1}{3}\right)^{-2}$

   • Using the property $a^{-n} = \frac{1}{a^n}$, we get: $\left(\frac{1}{3}\right)^{-2} = \frac{1}{\left(\frac{1}{3}\right)^2} = \frac{1}{\frac{1}{9}} = 9$

Step 2: Substitute and simplify

Now that we have evaluated each term, substitute back into the expression:

$4 - 9 = -5$

Final Answer

The result is: $-5$

Would you like further details on any part of this solution?

Here are some related questions that expand on this topic:

1. How do negative exponents work for different types of numbers?
2. What are the general rules for simplifying expressions with exponents?
3. Can we apply these exponent rules to variables as well as numbers?
4. How would the result change if the exponents were positive instead of negative?
5. What happens if we have fractional exponents?

Tip: Remember, a negative exponent flips the base; $(\frac{a}{b})^{-n}$ becomes $(\frac{b}{a})^n$.