小试牛刀数学已知a^2+3a+1=0.求a^3+1/a^3和a^7+1/a^7的值.

方程a^2+3a+1=0中,显然a≠0.
两边同时除以a,得:a+3+1/a=0,a+1/a= -3.
(a+1/a)²=(-3)^2,即a^2+1/a^2+2=9,a^2+1/a^2=7.
∴a^3+1/a^3=(a+1/a)(a^2+1/a^2)-(a+1/a)=(-3)*7-(-3)=-18;
(a^2+1/a^2)^2=7^2,即a^4+1/a^4+2=49,a^4+1/a^4=47.
∴a^7+1/a^7=(a^3+1/a^3)(a^4+1/a^4)-(a+1/a)=(-18)*47-(-3)=-843.

由a^2+3a+1=0，a+1/a=-3
a^2+1/a^2=(a+1/a)^2-2=7
a^3+1/a^3=(a+1/a)(a^2-1+1/a^2)=-3×6=-18
a^4+1/a^4=(a^2+1/a^2)^2-2=47
a^7+1/a^7=(a^3+1/a^3)(a^4+1/a^4)-(a+1/a)=-18×47+3=-843