The given expression is:

$2 \cdot 9^{-2} \div 9^{-4}$

Step-by-step Solution:

1. Apply the division of exponents rule: $a^m \div a^n = a^{m-n}$ Here, we are dividing $9^{-2}$ by $9^{-4}$, so:

   $9^{-2} \div 9^{-4} = 9^{-2 - (-4)} = 9^{-2 + 4} = 9^2$

2. Simplify the expression: Now the expression becomes: $2 \cdot 9^2$

3. Calculate $9^2$: $9^2 = 81$

4. Final multiplication: $2 \cdot 81 = 162$

So, the value of the expression is:

$\boxed{162}$

Do you have any questions or would you like further clarification?

Here are 5 related questions for practice:

1. Simplify $3 \cdot 5^{-1} \div 5^{-3}$.
2. What is the value of $2^4 \cdot 2^{-6}$?
3. Simplify $10^{-2} \times 10^5$.
4. How would you evaluate $8 \cdot 4^{-3} \div 4^{-1}$?
5. Solve $6 \cdot 2^3 \div 2^5$.

Tip:

When dividing powers with the same base, subtract the exponents. When multiplying powers, add the exponents.