Question

求解数学题！！！

已知：
a2/3+b2/3=4
x=a+3a1/3×b2/3
y=b+3a2/3×b1/3
求：（x+y）2/3－（x-y）2/3=？
题里的1/3和2/3都是前面字母的指数 这里打不出来上标
感谢各位高手哦

Answer 1

(x+y)^(2/3)-(x-y)^(2/3),是不是(x+y)^(2/3)+(x-y)^(2/3)？
如果是加，可以做

x+y=a+b+3a^(1/3)b^(2/3)+a^(2/3)b^(1/3)
=[a^(1/3)]^3+[b^(1/3)]^3+3a^(1/3)b^(2/3)+3a^(2/3)b^(1/3)
=[a^(1/3)+b^(1/3)][a^(2/3)-a^(1/3)b^(1/3)+b^(2/3)]+3a^(1/3)b^(1/3)[a^(1/3)+b^(1/3)]
=[a^(1/3)+b^(1/3)][a^(2/3)-a^(1/3)b^(1/3)+b^(2/3)+3a^(1/3)b^(1/3)]
=[a^(1/3)+b^(1/3)][a^(2/3)+2a^(1/3)b^(1/3)+b^(2/3)]
=[a^(1/3)+b^(1/3)]^3

x-y=a-b+3a^(1/3)b^(2/3)-a^(2/3)b^(1/3)
=[a^(1/3)]^3-[b^(1/3)]^3+3a^(1/3)b^(2/3)-3a^(2/3)b^(1/3)
=[a^(1/3)-b^(1/3)][a^(2/3)+a^(1/3)b^(1/3)+b^(2/3)]-3a^(1/3)b^(1/3)[a^(1/3)-b^(1/3)]
=[a^(1/3)-b^(1/3)][a^(2/3)+a^(1/3)b^(1/3)+b^(2/3)-3a^(1/3)b^(1/3)]
=[a^(1/3)-b^(1/3)][a^(2/3)-2a^(1/3)b^(1/3)+b^(2/3)]
=[a^(1/3)-b^(1/3)]^3

所以(x+y)^(1/3)=a^(1/3)+b^(1/3)
(x-y)^(1/3)=a^(1/3)-b^(1/3)
令m=a^(1/3),n=b^(1/3)
则(x+y)^(2/3)+(x-y)^(2/3)
=(m+n)^2+(m-n)^2
=2(m^2+n^2)
=2[a^(2/3)+b^(2/3)]
=2*4
=8

Answer 2

设：m=a^1/3, n=b^1/3

则：
m^2+n^2 = 4
而由：
x=m^3+3mn^2
y=n^3+3nm^2
可得
x+y=(m+n)^3
x-y=(m-n)^3
所以
（x+y）2/3－（x-y）2/3={（m+n)^3}^2/3 - {（m-n)^3}^2/3
= (m+n)^2-(m-n)^2
= 4mn = 4（ab）^1/3

无解，错题。

最后如果是两式相加，(m+n)^2+(m-n)^2的话有解，= 2(m^2+n^2)=8

P.S.:已知x^2+y^2 ,欲求xy的值， 是不可能的。

Answer 3

x+y=a+b+3a^(1/3)b^(2/3)+a^(2/3)b^(1/3)
=[a^(1/3)]^3+[b^(1/3)]^3+3a^(1/3)b^(2/3)+3a^(2/3)b^(1/3)
=[a^(1/3)+b^(1/3)][a^(2/3)-a^(1/3)b^(1/3)+b^(2/3)]+3a^(1/3)b^(1/3)[a^(1/3)+b^(1/3)]
=[a^(1/3)+b^(1/3)][a^(2/3)-a^(1/3)b^(1/3)+b^(2/3)+3a^(1/3)b^(1/3)]
=[a^(1/3)+b^(1/3)][a^(2/3)+2a^(1/3)b^(1/3)+b^(2/3)]
=[a^(1/3)+b^(1/3)]^3

x-y=a-b+3a^(1/3)b^(2/3)-a^(2/3)b^(1/3)
=[a^(1/3)]^3-[b^(1/3)]^3+3a^(1/3)b^(2/3)-3a^(2/3)b^(1/3)
=[a^(1/3)-b^(1/3)][a^(2/3)+a^(1/3)b^(1/3)+b^(2/3)]-3a^(1/3)b^(1/3)[a^(1/3)-b^(1/3)]
=[a^(1/3)-b^(1/3)][a^(2/3)+a^(1/3)b^(1/3)+b^(2/3)-3a^(1/3)b^(1/3)]
=[a^(1/3)-b^(1/3)][a^(2/3)-2a^(1/3)b^(1/3)+b^(2/3)]
=[a^(1/3)-b^(1/3)]^3

所以(x+y)^(1/3)=a^(1/3)+b^(1/3)
(x-y)^(1/3)=a^(1/3)-b^(1/3)
令m=a^(1/3),n=b^(1/3)
则(x+y)^(2/3)+(x-y)^(2/3)
=(m+n)^2+(m-n)^2
=2(m^2+n^2)
=2[a^(2/3)+b^(2/3)]
=2*4
=8

Answer 4

无解，错题。