https://file.aiquickdraw.com/mathbot/file/1736993841801sk3fxvb3.jpg

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!

Discover how to simplify the expression 2^(-5) ÷ (2^3 ÷ 2^6) using the properties of exponents. Learn key mathematical concepts such as handling negative exponents and simplifying division within parentheses. Suitable for Grades 8-10.

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