Let's break down and simplify the given expression step by step:

The expression is:

\[ 6^2 \cdot 2^{-5} \div 12^{-2} \]

1. **Calculate \( 6^2 \):**
   \[ 6^2 = 36 \]

2. **Calculate \( 2^{-5} \):**
   \[ 2^{-5} = \frac{1}{2^5} = \frac{1}{32} \]

3. **Calculate \( 12^{-2} \):**
   \[ 12^{-2} = \frac{1}{12^2} = \frac{1}{144} \]

Now the expression becomes:

\[ 36 \cdot \frac{1}{32} \div \frac{1}{144} \]

4. **Simplify the division:**
   Dividing by a fraction is the same as multiplying by its reciprocal:
   \[ 36 \cdot \frac{1}{32} \cdot 144 \]

5. **Perform the multiplications:**
   \[ 36 \cdot \frac{1}{32} = \frac{36}{32} = \frac{9}{8} \]
   \[ \frac{9}{8} \cdot 144 = \frac{9 \cdot 144}{8} = \frac{1296}{8} = 162 \]

So, the simplified value of the expression \( 6^2 \cdot 2^{-5} \div 12^{-2} \) is:

\[ 162 \]

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 negative exponents?
2. What is the rule for multiplying and dividing powers of different bases?
3. How can you simplify complex fractions?
4. What is the general rule for raising a number to a negative power?
5. How does the order of operations apply to exponentiation and multiplication?
6. What are some practical applications of negative exponents?
7. How do you simplify expressions involving multiple exponents?
8. Can you explain the concept of reciprocals and their role in division?

**Tip:** Always remember to follow the order of operations (PEMDAS/BODMAS) when simplifying expressions, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Learn how to simplify the expression 6 squared times 2 to the power of negative 5 divided by 12 to the power of negative 2 step-by-step. This problem involves exponents, negative exponents, and follows the order of operations. Suitable for Grades 7-9.

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