Let's solve the given expression step by step.

The expression is:
$2^{-5} \div \left(2^3 \div 2^6\right)$

Step 1: Simplify the parentheses $(2^3 \div 2^6)$

Using the property of exponents $a^m \div a^n = a^{m-n}$, we get:
$2^3 \div 2^6 = 2^{3-6} = 2^{-3}$

Step 2: Substitute back into the original expression

The expression becomes:
$2^{-5} \div 2^{-3}$

Step 3: Simplify the division

Again using the property $a^m \div a^n = a^{m-n}$, we get:
$2^{-5} \div 2^{-3} = 2^{-5 - (-3)} = 2^{-5 + 3} = 2^{-2}$

Step 4: Evaluate $2^{-2}$

The negative exponent rule states $a^{-n} = \frac{1}{a^n}$, so:
$2^{-2} = \frac{1}{2^2} = \frac{1}{4}$

Final Answer:

$\boxed{\frac{1}{4}}$

Would you like a deeper explanation or have any questions?

Here are 5 related questions to deepen your understanding:

1. What happens when you multiply powers with the same base, such as $2^m \cdot 2^n$?
2. How do you simplify $2^{-4} \cdot 2^2$?
3. What is the general rule for handling negative exponents?
4. How would the calculation differ if the base were $3$ instead of $2$?
5. Can you explain the difference between $(2^3)^2$ and $2^{3+2}$?

Tip: Always handle operations inside parentheses first and use exponent rules systematically!