The problem asks you to evaluate the following expression:

$\frac{5^9 \times 64 \times 3^5}{3^3 \times 2^4 \times 125} + 3^2$

Step-by-Step Solution:

1. Simplify powers and constants:

   • $5^9$ is a large number, so let's keep it as is for now.
   • $64 = 2^6$, so substitute that in.
   • $3^5 = 243$.
   • $3^3 = 27$.
   • $2^4 = 16$.
   • $125 = 5^3$.

2. Substitute values: $\frac{5^9 \times 2^6 \times 3^5}{3^3 \times 2^4 \times 5^3} + 3^2$

3. Cancel out the common terms:

   • In the numerator, we have $5^9$ and $5^3$, so $5^{9-3} = 5^6$.
   • In the numerator, we have $2^6$ and $2^4$, so $2^{6-4} = 2^2$.
   • In the numerator, we have $3^5$ and $3^3$, so $3^{5-3} = 3^2$.

   After simplifying, the expression becomes: $5^6 \times 2^2 \times 3^2 + 3^2$

4. Now calculate the remaining terms:

   • $5^6 = 15625$.
   • $2^2 = 4$.
   • $3^2 = 9$.

   So the expression is now: $15625 \times 4 \times 9 + 9$

5. Perform the multiplications: $15625 \times 4 = 62500$ $62500 \times 9 = 562500$

6. Add the final term: $562500 + 9 = 562509$

Final Answer:

$\boxed{562509}$

Would you like more details on any part of this process, or have any further questions? Here are a few related questions for you to explore:

1. How can we break down large exponents using properties of exponents?
2. What are some shortcuts for simplifying large expressions involving powers of numbers?
3. How does factoring help in simplifying expressions with powers?
4. What is the significance of canceling common terms in a fraction?
5. Can you explain the difference between multiplying and dividing exponents with the same base?

Tip: When dealing with large exponents, breaking them down into smaller powers can make calculations easier.