已知a^2-a-1=0,求a^18+323a^-6的值

a^2-a-1=0
a^2=a+1
a^4=(a^2)^2=a^2+2a+1=(a+1)+2a+1=3a+2
a^8=(a^4)^2=(3a+2)^2=9a^2+12a+4=9(a+1)+12a+4=21a+13
a^16=(a^8)^2=(21a+13)^2=441a^2+546a+169=441(a+1)+546a+169=987a+610
a^24=a^8*a^16=(21a+13)(987a+610)=20727a^2+25641a+7930=20727(a+1)+25641a+7930=46368a+28657
a^6=a^2*a^4=(a+1)(3a+2)=3a^2+5a+2=3(a+1)+5a+2=8a+5
所以a^18+323a^-6
=a^18+323/a^6
=(a^24+323)/a^6
=(46368a+28657+323)/(8a+5)
=5796(8a+5)/(8a+5)
=5796

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a^2-a-1=0
除以a，再移项，有：a-a^(-1)=1
平方，有：a^2+a^(-2)-2=1
所以：a^3-a^(-3) =(a-a^(-1))*(a^2+a^(-2)+1) =4
平方，有：a^6+a^(-6)=18
平方：a^12+a^(-12)=322
a^18+323×a^(-6) =a^18+a^(-6)+322*a^(-6)
=a^6*(a^12+a^(-12))+322*a^(-6)
=322*a^6*+322*a^(-6)
=322*(a^6*+a^(-6))
=322*18
=5796