# x^1/3 + y^1/3 + z^1/3 = 0find (x + y + z)^3

$\\Let x^{1/3} = a, y^{1/3} = b, z^{1/3} = c \\The question then becomes, if (a+b+c) = 0 \\ then what is the value of (a^3+b^3+c^3)^3. \\We have a+b = -c. \\=> (a+b)^3 = (-c)^3 = -c^3. \\=> a^3+b^3+3*a*(b^2)+3*(a^2)*b = -c^3 \\=> a^3+b^3+c^3 = -{ 3*a*(b^2)+3*(a^2)*b } = -3*a*b*(a+b) = -3*a*b*(-c) = 3abc. \\=> (a^3+b^3+c^3)^3 = (3abc)^3 = 27(a^3)(b^3)(c^3) = 27xyz.$