Sum Real x: Solving ((2^x)-4)^3 + ((4^x)-2)^3

[ritwik06]

Homework Statement

Find the sum of all real x such that

((2^x)-4)^3 + ((4^x)-2)^3=((4^x)+(2^3)-6)^3

Homework Equations

Nil

The Attempt at a Solution

(a-b)^3+3ab(a-b)=a^3-b^3

This equation doesn't help as the LHS is a sum of two cubes.

Is it wise to convert all fours and sixes as multiples of 2?

Please help me to get a start at least.

 

[HallsofIvy]

It is also true that a^3+ b^3= (a+ b)(a^2- ab+ b^2).