First, we notice that both sides of the equation (2w+8)^(1/3) = (8w-4)^(1/3) have the same exponent of 1/3. Therefore, we can cube both sides to get rid of the cube root.
This gives us the equation (2w+8) = (8w-4).
To simplify the equation further and solve for w, we can start by moving the terms on the left to the right and the terms on the right to the left, in order to gather like terms. This gives us:
2w - 8w = -4 - 8
Simplifying that further, we obtain the equation:
-6w = -12
Then, we divide both sides of the equation by -6 to solve for w:
w = -12 / -6
Therefore, the solution to the equation is:
w = 2
So, the real number 'w' that satisfies the initial equation (2w+8)^(1/3) = (8w-4)^(1/3) is w = 2.