Q: If x^1/3 + y^1/3 + z^1/3 = 0 , then the value of ( x+y+z )^3 is :
a) 27
b) 27xyz
c) 81
d)(xyz)^3
e)xyz
a) 27
b) 27xyz
c) 81
d)(xyz)^3
e)xyz
800. Arjun's-Bird-Eye
Let x^1/3 = a, y^1/3 = b, z^1/3 = c
The question then becomes, if (a+b+c) = 0, then what is the value of (a^3+b^3+c^3)^3.
We have a+b = -c.
Or (a+b)^3 = (-c)^3 = -c^3.
Or a^3+b^3+3*a*(b^2)+3*(a^2)*b = -c^3
Or a^3+b^3+c^3 = -{ 3*a*(b^2)+3*(a^2)*b } = -3*a*b*(a+b) = -3*a*b*(-c) = 3abc.
So (a^3+b^3+c^3)^3 = (3abc)^3 = 27(a^3)(b^3)(c^3) = 27xyz.
The correct answer is (b).
