If a^1/3 + b^1/3 + c^1/3 = 0, then a relation among a, b, c is?

Answer: Option B

$$\begin{array}{rl}& a^{\frac{1}{3}}+b^{\frac{1}{3}}+c^{\frac{1}{3}}=0\\ & \Rightarrow a^{\frac{1}{3}}+b^{\frac{1}{3}}=-c^{\frac{1}{3}}\\ & \text{Take cube on both sides}\\ & \Rightarrow {\left(a^{\frac{1}{3}}+b^{\frac{1}{3}}\right)}^3={\left(-c^{\frac{1}{3}}\right)}^3\\ & \Rightarrow a+b+3a^{\frac{1}{3}}{b}^{\frac{1}{3}}\left(a^{\frac{1}{3}}+b^{\frac{1}{3}}\right)=-c\\ & \Rightarrow a+b+3a^{\frac{1}{3}}{b}^{\frac{1}{3}}\left(-c^{\frac{1}{3}}\right)=-c\\ & \Rightarrow a+b+c=3a^{\frac{1}{3}}{b}^{\frac{1}{3}}{c}^{\frac{1}{3}}\\ & \text{Again taking cube }\\ & \Rightarrow {\left(a+b+c\right)}^3=27abc\end{array}$$