The expression you provided is:

$\frac{7^5 \cdot 7^6}{7^3 \cdot 7^{-2}}$

Step-by-step solution:

1. Simplify the numerator: When multiplying powers of the same base, you add the exponents. $7^5 \cdot 7^6 = 7^{5 + 6} = 7^{11}$

2. Simplify the denominator: Similarly, for the denominator: $7^3 \cdot 7^{-2} = 7^{3 + (-2)} = 7^1$

3. Simplify the fraction: When dividing powers of the same base, subtract the exponents: $\frac{7^{11}}{7^1} = 7^{11 - 1} = 7^{10}$

Final answer:

$7^{10}$

Would you like any more details or further explanation?

Here are some follow-up questions you might find interesting:

1. How do you simplify expressions with powers that have different bases?
2. What is the general rule for multiplying powers of the same base?
3. How do you handle negative exponents when simplifying an expression?
4. Can you think of a real-world scenario where powers of numbers might be useful?
5. How do we simplify powers when both multiplication and division are involved?

Tip: Always remember the rules for handling exponents (addition, subtraction, multiplication) are only valid when the bases are the same!