To evaluate the expression
\[ \left( \frac{6^7 \cdot 3^3}{6^6 \cdot 3^4} \right)^3, \]
we can simplify the fraction inside the parentheses first.
- Simplify the fraction \(\frac{6^7 \cdot 3^3}{6^6 \cdot 3^4}\):
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
\[ \frac{6^7}{6^6} = 6^{7-6} = 6^1 = 6 \]
And for the powers of 3:
\[ \frac{3^3}{3^4} = 3^{3-4} = 3^{-1} = \frac{1}{3} \]
Now combine these results:
\[ \frac{6^7 \cdot 3^3}{6^6 \cdot 3^4} = 6 \cdot \frac{1}{3} = \frac{6}{3} = 2 \]
- Now cube the result:
\[ \left(2\right)^3 = 2^3 = 8 \]
Thus, the value of the entire expression
\[ \left( \frac{6^7 \cdot 3^3}{6^6 \cdot 3^4} \right)^3 = 8. \]
So, the final answer is
\[ \boxed{8}. \]