https://file.aiquickdraw.com/mathbot/file/1725407272313ge5lv89a.jpg

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?

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**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.

Simplify the expression \( \frac{9^{-2} \cdot 9^7}{9^3} \).

This step-by-step solution explains how to simplify the expression \( \frac{9^{-2} \cdot 9^7}{9^3} \) using properties of exponents. Ideal for students in Grades 8-10, it covers key concepts in algebra and exponentiation.

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