Question

x = 2^1/3 + 2^-1/3 prove 2x^3 = 6x + 5

Answer 1

Let the cube root of 2 be y.
=>x = y + 1/y
=>2x^3 = 2(y + 1/y)^3
             = 2(y^3 + 1/y^3 + 3y + 3/y)
             = 2(y^3 + 1/y^3) + 6(y + 1/y)

Note that y^3 = 2.

=> 2x^3 = 2(2 + 1/2) + 6x                              (y + 1/y = x as mentioned above)
              = 5 + 6x