2011/104) If a=(root 2+1)^ 1/3 - (root 2 -1 )^1/3 then value of a ^ 3 +3a-2 is zero. explain

a = -(root 2+1)^ 1/3-(root 2 -1)^ 1/3

so a + (-(root 2+1)^ 1/3) +(root 2 -1)^ 1/3 = 0

using x+y+z= 0 => x^3 +y^3 + z^3 = 3xyz

we get

a^3 + (-(root 2+1)) + ((root 2-1)) = 3a((-(root 2+1))((root 2-1)

or a^3 - 2 = - 3a and hence a^3+3a - 2 = 0