若x>0,y>0,且√(x^2+³√x^4y^2)+√(y^2+³√y^4x^2)=8求x^2/3+y^2/3的值.
问题描述：
若x>0,y>0,且√(x^2+³√x^4y^2)+√(y^2+³√y^4x^2)=8求x^2/3+y^2/3的值.
答
令x^(2/3)=a ,y^(2/3)=b
√(x^2+³√x^4y^2)+√(y^2+³√y^4x^2)
=[x²+x^(4/3)y^(2/3)^(1/2)]+[y²+y^(4/3)x^(2/3]^(1/2)
=(a³+a²b)^(1/2)+(b³+b²a)^(1/2)
=a(a+b)^(1/2)+b(b+a)^(1/2)
=(a+b)^(1/2)×(a+b)
=(a+b)^(3/2)
8=4^(3/2)
(a+b)^(3/2)=4^(3/2)
a+b=4
所以,x^(2/3)+y^(2/3)=4