jmc

By Vieta's formulas, $r+s=\sqrt{5}$ and $rs=1.$ Squaring the equation $r+s=\sqrt{5},$ we get $$r^2+2rs+s^2=5,$$so $r^2+s^2=5-2rs=3.$ Squaring this equation, we get $$r^4+2r^2{s}^2+s^4=9,$$so $r^4+s^4=9-2r^2{s}^2=9-2=7.$ Squaring once more, we get $$r^8+2r^4{s}^4+s^8=49,$$so $r^8+s^8=49-2r^4{s}^4=\boxed{47}.$