jmc

Multiplying both sides by $rs(r+s),$ we get $$rs=s(r+s)+r(r+s).$$This simplifies to $r^2+rs+s^2=0.$ Then $$(r-s)(r^2+rs+s^2)=0,$$which expands as $r^3-s^3=0.$ Therefore, ${\left(\frac{r}{s}\right)}^3=\boxed{1}.$