Overview

Let $s$ be the length of each edge (side) of the cube:

1. Relate volume $V$ to edge length $s$: $V = s^3 \implies s = \sqrt[3]{V} = V^{1/3}$

2. Relate total surface area $S$ to edge length $s$: A cube has 6 identical square faces, each of area $s^2$: $S = 6s^2$

3. Substitute $s = V^{1/3}$ to express $S$ as a function of $V$: $S(V) = 6\left(V^{1/3}\right)^2 = 6V^{2/3}$