Consider $R=\frac {19^{17}}{17^{19}}$
$R=\frac {19^{17}}{17^{17}17^{2}}=\frac 1{289} \frac {19^{17}}{17^{17}}$
$R=\frac 1{289}(1+ \frac {2}{17})^{17}$
It is clear that $\frac {2}{17}=\frac {6}{51}<\frac {6}{50} \Rightarrow (1+\frac {2}{17})<(1+\frac {6}{50})$
So $R<\frac 1{289}(1+ \frac {3}{25})^{17}$
$R<\frac 1{289}(1 + 17 \frac {3}{25} + {{16.17} \over 2} \frac {3}{25}\frac {3}{25}+{{15.16.17} \over {2.3}} \frac {3}{25}\frac {3}{25}\frac {3}{25} + ... )$
$R<\frac 1{289}(1 + \frac {17.3}{25} + {{17.3} \over 25} (\frac {16}{2}\frac {3}{25})+{{17.3} \over 25} (\frac {16}{2}\frac {3}{25})(\frac {15}{3}\frac {3}{25})+{{17.3} \over 25} (\frac {16}{2}\frac {3}{25})(\frac {15}{3}\frac {3}{25})(\frac {14}{4}\frac {3}{25}) + ... )$
Each successive term is now being multiplied by something like $(\frac {18-k}{k}\frac {3}{25})<(\frac {15}{3}\frac {3}{25})=\frac {3}{5}$
So $R<\frac 1{289}(1 + \frac {17.3}{25} + {{17.3} \over 25} (\frac {16}{2}\frac {3}{25})+{{17.3} \over 25} (\frac {16}{2}\frac {3}{25})(\frac {3}{5})+{{17.3} \over 25} (\frac {16}{2}\frac {3}{25})(\frac {3}{5})^2 + {{17.3} \over 25} (\frac {16}{2}\frac {3}{25})(\frac {3}{5})^3... )$
Sum of infinite geometric sequence gives us
So $R<\frac 1{289}(1 + \frac {17.3}{25} + {{17.3} \over 25} (\frac {16}{2}\frac {3}{25})(\frac {5}{2}) )$
So $R<\frac {7.936}{289}$
Thus $R<1$
$\frac {19^{17}}{17^{19}}<1$
${19^{17}}<{17^{19}}$