若(x-1)(y+1)=3,xy(x-y)=4，求：(1)x^2+y^2;(2)x^3-y^3;(3)x^4+y^4;(4)x^7-y^7

如题所述

(x-1)(y+1)=3
xy+x-y=4
解得,x-y=4-xy
代入:xy(x-y)=4
xy(4-xy)=4
4xy-(xy)²-4=0
x²y²-4xy+4=0
(xy-2)²=0
解得,xy=2
x-y=4-2=2

(1)x^2+y^2
=(x-y)²+2xy
=2²+4
=8

(2)x^3-y^3
=(x-y)(x²+xy+y²)
=2×(8+2)
=20

(3)x^4+y^4
=(x²+y²)²-2(xy)²
=8²-2×2²
=56

(4)x^7-y^7
=(x^4+y^4)(x³-y³)+x^4y³-x³y^4
=56×20+x³y³(x-y)
=56×20+2³×2
=1136

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