The expression in the image is:

$\frac{9^{-2} \cdot 9^7}{9^3}$

To simplify this expression, we can use the properties of exponents. Specifically:

1. Multiplying Powers with the Same Base: When multiplying powers with the same base, you add the exponents: $a^m \cdot a^n = a^{m+n}$.

2. Dividing Powers with the Same Base: When dividing powers with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Step-by-Step Solution:

1. Combine the exponents in the numerator:

   $9^{-2} \cdot 9^7 = 9^{-2 + 7} = 9^5$

2. Now simplify the expression by subtracting the exponent in the denominator:

   $\frac{9^5}{9^3} = 9^{5-3} = 9^2$

3. Final Answer:

   $9^2 = 81$

So, the simplified result is 81.

Would you like more details or have any questions?

Related Questions:

1. What is the general rule for multiplying and dividing exponents with the same base?
2. How would the expression change if the base were different, for example, $7$ instead of $9$?
3. Can you solve a similar expression with different exponents, such as $\frac{4^{-3} \cdot 4^5}{4^2}$?
4. What happens if the exponents are fractions instead of integers?
5. How would you simplify $\frac{2^{-1} \cdot 2^3 \cdot 2^{-2}}{2^2}$?

Tip: Always remember to handle the exponents first before performing any other operations when simplifying expressions.