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).

 

[Architeuthis Dux]

I would first clarify what the goal of the problem is. Is the goal to find the specific values of x that satisfy the given equation? Or is it to find the sum of all real x that satisfy the equation? This information is important in determining the approach to solving the problem.

If the goal is to find the specific values of x, then we can start by simplifying the equation using the given equation (a-b)^3+3ab(a-b)=a^3-b^3. However, as you mentioned, this may not be very useful since the LHS is a sum of two cubes.

If the goal is to find the sum of all real x, then we can use the fact that the sum of two cubes is equal to the cube of their sum minus three times the product of the cubes. This means that we can rewrite the equation as ((2^x)+(4^x)-6)^3=((4^x)+(2^3)-6)^3. This allows us to cancel out the cubes on both sides, leaving us with (2^x)+(4^x)-6=(4^x)+(2^3)-6. Simplifying this further, we get (2^x)+(4^x)=(4^x)+(2^3). This can be rewritten as (2^x)+(2^x)^2=(2^x)^2+(2^3).

From here, we can see that the only value of x that satisfies this equation is x=0. Therefore, the sum of all real x that satisfy the given equation is 0.

In conclusion, the approach to solving this problem depends on the goal of the problem. If the goal is to find specific values of x, then simplifying the equation using the given equation (a-b)^3+3ab(a-b)=a^3-b^3 may be helpful. If the goal is to find the sum of all real x, then using the fact that the sum of two cubes is equal to the cube of their sum minus three times the product of the cubes can be a useful approach.