已知x-y=1,x^2+y^2=3,求x^6-y^6的值

已知x-y=1,x^2+y^2=3,求x^6-y^6的值
已知x-y=1,x^2+y^2=3,求x^6-y^6的值

已知x-y=1,x^2+y^2=3,求x^6-y^6的值

说明：本题主要是对平方和及平方差公式的灵活应用.

x-y=1

(x-y)^2=1^2

x^2+y^2-2xy=1

3-2xy=1

xy=1

(x+y)^2=x^2+y^2+2xy=3+2=5

所以x+y=±√5

于是

x^6-y^6
=(x^3-y^3)(x^3+y^3)
=(x-y)(x^2+xy+y^2)(x+y)(x^2-xy+y^2)

=1*(3+1)*(±√5)*(3-1)

=4*(±√5)*2

=±8√5

答：
x-y=1
x^2+y^2=3
所以：(x-y)^2=x^2-2xy+y^2=3-2xy=1
解得：xy=1
所以：(x+y)^2=x^2+2xy+y^2=3+2=5
解得：x+y=±√5
x^6-y^6
=(x^3-y^3)(x^3+y^3)
=(x-y)(x+y)(x^2+xy+y^2)(x^2-xy+y^2)
=1×(±√5)×(3+1)×(3-1)
=±8√5