1,（1）已知a＋b＝1,a²＋b²＝2.求a的七次方与b的七次方的和.

1,（1）已知a＋b＝1,a²＋b²＝2.求a的七次方与b的七次方的和.
a+b=1
（a+b）^2=a^2+2ab+b^2=1
又:a^2+b^2=2
2ab=-1
ab=-1/2
(a+b)(a^2+b^2)=a^3+ab^2+ba^2+b^3=a^3+b^3+ab(a+b)=1*2=2
a^3+b^3=2-ab(a+b)=2-(-1/2)*1=2.5
(a^3+b^3)(a^2+b^2)=a^5+a^2b^2(a+b)+b^5=2.5*2=5
a^5+b^5=5-(ab)^2(a+b)=5-(-1/2)^2=19/4
(a^5+b^5)(a^2+b^2)=a^7+b^7+a^2b^5+b^2a^5=a^7+b^7+a^2b^2(a^3+b^3)=19/4*2=19/2
a^7+b^7=19/2-a^2b^2(a^3+b^3)=19/2-(-1/2)^2*2.5=19/2-2.5/4=35.5/4=71/8 .
（2）已知х－у＝1,х³－у³＝2.求х的4次方与у的4次方的和及х的5次方与у的5次方的差.
x^3-y^3=(x-y)(x^2+xy+y^2)=2
x^2+xy+y^2=2
又 x^2-2xy+y^2=1
3xy=1
xy=1/3
x^2+y^2=5/3
(x^2+y^2)^2=x^4+y^4+2x^2y^2=25/9
x^4+y^4=25/9-2(xy)^2=25/9-2(1/3)^2=23/9
(x^3-y^3)(x^2+y^2)=2*5/3=10/3
x^5-y^5-x^2y^3+x^3y^2=x^5-y^5+x^2y^2(x-y)=10/3
x^5-y^5=10/3-x^2y^2(x-y)=10/3-(1/3)^2=10/3-1/9=29/9
即：x^5-y^5=29/9 .