The shortest path from P to Q along the surface of the cube will be one that goes from P, to the center point on any edge, to the opposite side, and then to Q. The length of that will be L + L/2 + L/2, or 2L, where L is the length of any side of the cube.
We are told that this distance is equal to 2√2, so we can say:
2L = 2√2 cm
∴ L = √2 cm
The volume of the cube will be that length, cubed, or L3, which means we can say:
V = L3
∴ V = (√2 cm)3
∴ V = (21/2)3 cm3
∴ V = 23/2 cm3