If $$\frac{1}{\sqrt[3]{4}+ \sqrt[3]{2}+1}= a^{\sqrt[3]{4}}+b^{\sqrt[3]{2}}+c$$ and a, b, c are rational numbers. then a + b + c is equal to

Expression : $$\frac{1}{\sqrt[3]{4}+ \sqrt[3]{2}+1}= a^{\sqrt[3]{4}}+b^{\sqrt[3]{2}}+c$$

=> $$\frac{1}{2^\frac{2}{3} + 2^\frac{1}{3} + 1} = a.2^\frac{2}{3} + b.2^\frac{1}{3} + c$$

=> $$\frac{2^\frac{1}{3} - 1}{(2^\frac{1}{3} - 1) (2^\frac{2}{3} + 2^\frac{1}{3} + 1)} = a.2^\frac{2}{3} + b.2^\frac{1}{3} + c$$

=> $$\frac{2^\frac{1}{3} - 1}{2 - 1} = a.2^\frac{2}{3} + b.2^\frac{1}{3} + c$$

=> Comparing both sides $$a = 0 , b = 1 , c = -1$$

To find : $$a + b + c$$

= $$0 + 1 - 1 = 0$$