The expression provided in the image is:

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

Step-by-Step Solution:

1. Simplify the numerator:

   • Using the law of exponents $a^m \cdot a^n = a^{m+n}$, combine the terms in the numerator: $6^3 \cdot 6^2 = 6^{3+2} = 6^5$

2. Simplify the denominator:

   • Apply the power rule $(a^m)^n = a^{m \cdot n}$ to simplify the denominator: $(6^2)^4 = 6^{2 \cdot 4} = 6^8$

3. Combine the fraction:

   • Now, rewrite the fraction as: $\frac{6^5}{6^8}$
   • Using the quotient rule $\frac{a^m}{a^n} = a^{m-n}$, subtract the exponents: $6^{5-8} = 6^{-3}$

4. Final Answer: $6^{-3}$

   This can be expressed as: $\frac{1}{6^3} = \frac{1}{216}$

The simplified form of the expression is $\frac{1}{216}$.

Would you like further details or explanations?

Related Questions:

1. What is the general rule for multiplying exponents with the same base?
2. How do you simplify expressions with negative exponents?
3. Can you explain why $(a^m)^n = a^{m \cdot n}$?
4. How would you simplify a fraction with variables and exponents?
5. What is the result of $\frac{a^m}{a^n}$ if $m < n$?

Tip: Always apply the exponent rules step-by-step to avoid errors in simplification, especially when combining multiple terms with exponents.