jmc

First, we simplify a small portion of the question: $\frac{8}{9}\cdot {\left(\frac{1}{2}\right)}^4=\frac{2^3}{9}\cdot \frac{1^4}{2^4}$. Multiplying these two and combining numerator and denominator, we get $\frac{1\cdot 2^3}{9\cdot 2^4}$, which can be made into two fractions and simplified: $\frac{1}{9}\cdot \frac{2^3}{2^4}=\frac{1}{9}\cdot \frac{1}{2}=\frac{1}{18}$.

The inverse of this fraction is $\frac{1}{\frac{1}{18}}=18$, and $18-17=\boxed{1}$.