- 数学题已知a^2+b^2=5,x=a^3+3ab^2,y=b^3
- 已知a^2+b^2=5,x=a^3+3ab^2,y=b^3+3a^2b,求(x+y)^(2/3)+(x-y)^(2/3)的值。
- x+y=a^3+b^3+3ab(a+b)=(a+b)(a^2-ab+b^2)+3ab(a+b)
=(a+b)(a^2+2ab+b^2)=(a+b)^3
x-y=a^3-b^3-3ab(a-b)=(a-b)(a^2+ab+b^2)-3ab(a-b)
=(a-b)(a^2-2ab+b^2)=(a-b)^3
(x+y)^(2/3)+(x-y)^(2/3)
=[(a+b)^3]^(2/3)+[(a-b)^3]^(2/3)
=(a+b)^2+(a-b)^2
=2(a^2+b^2)=2*5=10