To solve the given mathematical expression:

$\frac{4^3 \cdot (7^2)^3}{(-3)^5} \div \left( \frac{(-4)^2 \cdot 7^5}{36} \right)$

We'll break down the operations step-by-step:

Step 1: Evaluate Powers and Squares

1. $4^3 = 64$
2. $7^2 = 49$ and $(7^2)^3 = 49^3 = 117649$
3. $(-3)^5 = -243$
4. $(-4)^2 = 16$

Step 2: Substitute and Simplify

• Replace the values in the expression: $\frac{64 \cdot 117649}{-243} \div \left( \frac{16 \cdot 16807}{36} \right)$

Step 3: Calculate the Numerators

1. $64 \cdot 117649 = 7529536$
2. $16 \cdot 16807 = 268912$

Step 4: Solve Each Fraction

1. $\frac{7529536}{-243} = -30989$ (rounded to nearest integer)
2. $\frac{268912}{36} = 7472$

Step 5: Divide the Results

$\frac{-30989}{7472} \approx -4.148$

The result of the expression is approximately $-4.148$.

Would you like more details on any of these steps, or do you have any further questions on this or another topic?

Here are some related questions you might find interesting:

1. How does the power of a negative number affect the result (odd vs. even powers)?
2. What are the rules for dealing with powers and roots in mathematical expressions?
3. Can you provide more examples of simplifying complex fractions?
4. How can exponent rules be used to simplify expressions involving variables?
5. What is the difference between direct division and inverse multiplication in expressions?

Tip: Remember that any negative number raised to an odd power will result negative, and when raised to an even power, it will result positive. This is crucial for correctly solving expressions with mixed signs.