a^2-a-1=0
a^2=a+1
a^2-a=1
a(a-1)=1
a-1=1/a=a^(-1)
a^8+7a^(-4)
=(a^2)^4+7*[a^(-1)]^4
=(a+1)^4+7*(a-1)^4
=(a^2+2a+1)^2+7*(a^2-2a+1)^2
=(a+1+2a+1)^2+7*(a+1-2a+1)^2
=(3a+2)^2+7*(a-2)^2
=9a^2+4+12a+7*(a^2+4-4a)
=16a^2-16a+32
=16*(a^2-a)+32
=16+32=48