jmc

We have $$\frac{1}{2}{x}^6{y}^7=\frac{1}{2}{\left(\frac{3}{4}\right)}^6{\left(\frac{4}{3}\right)}^7=\frac{1}{2}\cdot \frac{3^6}{4^6}\cdot \frac{4^7}{3^7}=\frac{1}{2}\cdot \frac{3^6}{3^7}\cdot \frac{4^7}{4^6}=\frac{1}{2}\cdot \frac{1}{3}\cdot 4=\boxed{\frac{2}{3}}.$$

We also could have tackled this problem quickly by noticing that if $x=\frac{3}{4}$ and $y=\frac{4}{3}$, then $xy=1$, so $\frac{1}{2}{x}^6{y}^7=\frac{1}{2}(xy{)}^{6}y=\frac{1}{2}\cdot 1^{6}y=\frac{1}{2}y=\frac{2}{3}$.