已知（x+y）^2=7 （x-y）^2=3计算x^6+y^6

题目

已知（x+y）^2=7 （x-y）^2=3计算x^6+y^6
已知（x+y）^2=7 （x-y）^2=3计算x^6+y^6

答案

(x+y)²=x²+2xy+y²=7
(x-y)²=x²-2xy+y²=3
两式相加,x²+y²=5
两式相减,xy=1
x^6+y^6
=(x²)³+(y²)³
=(x²+y²)(x^4-x²y²+y^4)
=5[(x²+y²)²-3xy]
=5×(25-3)
=110