To simplify the given expression using the method specified: $(2^3)^2 \cdot 2^{-5} / (2^6)^{-2}$, let's break it down step by step.

First, simplify each term:

1. Simplify $(2^3)^2$: $(2^3)^2 = 2^{3 \cdot 2} = 2^6$

2. Simplify $2^{-5}$: $2^{-5}$

3. Simplify $(2^6)^{-2}$: $(2^6)^{-2} = 2^{6 \cdot (-2)} = 2^{-12}$

Now the expression is: $2^6 \cdot 2^{-5} \div 2^{-12}$

Next, apply the properties of exponents:

4. Combine the terms $2^6$ and $2^{-5}$: $2^6 \cdot 2^{-5} = 2^{6-5} = 2^1 = 2$

5. Divide by $2^{-12}$: $2 \div 2^{-12} = 2 \cdot 2^{12} = 2^{1+12} = 2^{13}$

So, the simplified value of the expression $(2^3)^2 \cdot 2^{-5} / (2^6)^{-2}$ is: $2^{13}$

And $2^{13}$ equals: $2^{13} = 8192$

Thus, the simplified value of the expression is:

$8192$

Would you like any further details or have any questions?

Here are 8 related questions for further exploration:

1. How do you simplify expressions with exponents raised to other exponents?
2. What is the rule for multiplying powers with the same base?
3. What is the rule for dividing powers with the same base?
4. How can you simplify expressions with both positive and negative exponents?
5. What is the general rule for dealing with exponents in fraction form?
6. How do you handle multiple exponentiation operations in a single expression?
7. Can you explain why dividing by a negative exponent is equivalent to multiplication by its positive counterpart?
8. What are some real-world applications of exponential calculations?

Tip: When working with exponents, always use the properties of exponents to combine like terms before performing arithmetic operations.