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Let's simplify the expression step by step. The given expression is:

\[
\frac{6^3 \cdot 6^2}{(6^2)^4}
\]

### Step 1: Simplify the numerator
The numerator is \( 6^3 \cdot 6^2 \). Using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we can combine the powers of 6:
\[
6^3 \cdot 6^2 = 6^{3+2} = 6^5
\]

### Step 2: Simplify the denominator
The denominator is \( (6^2)^4 \). Using the power of a power rule \( (a^m)^n = a^{m \cdot n} \), we simplify:
\[
(6^2)^4 = 6^{2 \cdot 4} = 6^8
\]

### Step 3: Combine the numerator and denominator
Now, the expression becomes:
\[
\frac{6^5}{6^8}
\]

Using the quotient rule of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we get:
\[
\frac{6^5}{6^8} = 6^{5-8} = 6^{-3}
\]

### Final Answer:
The simplified expression is \( 6^{-3} \), which corresponds to **option A**.

Would you like further explanation on any of these steps?

### Follow-up questions:
1. How do you apply the product rule for exponents?
2. What is the power of a power rule, and how is it used here?
3. Can you explain the quotient rule for exponents in more detail?
4. How would the answer change if the base were different, say 7 instead of 6?
5. What happens if the exponents are fractions instead of integers?

**Tip:** When simplifying expressions with exponents, remember to handle the numerator and denominator separately before applying the quotient rule.

Which of the following simplifies the expression 6^3 * 6^2 / (6^2)^4?

This page provides a step-by-step guide to simplifying the expression 6^3 * 6^2 / (6^2)^4. Learn about the product, power, and quotient of powers rules for exponents, suitable for students in Grades 9-11.

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