已知x满足x²-x-1=0,求代数式x的八次方+7x的负四次方的值

已知x满足x²-x-1=0,求代数式x的八次方+7x的负四次方的值
已知x满足x²-x-1=0,求代数式x的八次方+7x的负四次方的值

已知x满足x²-x-1=0,求代数式x的八次方+7x的负四次方的值
x²-x=1
=>x²=x+1
=>x³=x²+x=2x+1
=>x^4=2x²+x=3x+2
=>x^5=3x²+2x=5x+3
=>x^6=5x²+3x=8x+5
=>x^7=8x²+5x=13x+8
=>x^8=13x²+8x=21x+13
=>x^8+7x^(-4)=21x+13+7/(3x+2)
=(63x²+81x+33)/(3x+2)
=(144x+96)/(3x+2)
=48(3x+2)/(3x+2)
=48