Let's simplify the expression step by step:
We start with:
\[ \frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}} \]
First, we simplify \((7^{-3})^7\):
\[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]
Now, we substitute this back into the expression:
\[ \frac{7^{-21} \cdot 7^{20}}{6^{-2}} \]
Next, we simplify the numerator \(7^{-21} \cdot 7^{20}\):
Using the property of exponents, \(a^m \cdot a^n = a^{m+n}\):
\[ 7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1} \]
Now our expression looks like this:
\[ \frac{7^{-1}}{6^{-2}} \]
Next, we can rewrite \(6^{-2}\) in the denominator:
\[ \frac{7^{-1}}{6^{-2}} = 7^{-1} \cdot 6^2 = \frac{1}{7} \cdot 36 = \frac{36}{7} \]
Thus, the final answer is:
\[ \boxed{\frac{36}{7}} \]