\[\frac{3^{6} \cdot 27}{81²} = \frac{3^{6} \cdot 3^{3}}{(3^{4})²} = \frac{3^{6 + 3}}{3^{4 \cdot 2}} = \frac{3^{9}}{3^{8}} = 3^{9 - 8} = 3.\]