jmc

Squaring the equation $4=a+a^{-1},$ we get $$16={\left(a+a^{-1}\right)}^2=a^2+2a{a}^{-1}+a^{-2}=a^2+2+a^{-2},$$so $14=a^2+a^{-2}.$ To get the desired expression, we square again, giving $$196=a^4+2a^2{a}^{-2}+a^{-4}=a^4+2+a^{-4}.$$Thus, $\boxed{194}=a^4+a^{-4}.$