已知a^(2/3)+b^(2/3)=4,x=a+3a^(1/3)*b^(2/3),y=b+3a^(2/3)*b^(1/3
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已知a^(2/3)+b^(2/3)=4,x=a+3a^(1/3)*b^(2/3),y=b+3a^(2/3)*b^(1/3),试证明:(x-y)^(2/3)+(x+y)^(2/3)的值与x,y的取值无关
x=a+3a^(1/3)*b^(2/3),y=b+3a^(2/3)*b^(1/3),
∴x-y=[a^(1/3)-b^(1/3)]^3,
x+y=[a^(1/3)+b^(1/3)]^3,
∴(x-y)^(2/3)+(x+y)^(2/3)
=[a^(1/3)-b^(1/3)]^2+[a^(1/3)+b^(1/3)]^2
=2[a^(2/3)+b^(2/3)]
=8.
∴x-y=[a^(1/3)-b^(1/3)]^3,
x+y=[a^(1/3)+b^(1/3)]^3,
∴(x-y)^(2/3)+(x+y)^(2/3)
=[a^(1/3)-b^(1/3)]^2+[a^(1/3)+b^(1/3)]^2
=2[a^(2/3)+b^(2/3)]
=8.
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