The answer is 1:64
Let's say that the original cube had side lengths of 1 unit. The volume would be 1*1*1 = 1 cubic unit.
If we quadruple each side, then the new cube has side lengths of 4. The new volume is 4*4*4 = 64
The old volume (1 cubic unit) pairs up with the new volume (64 cubic units) to form the ratio 1:64
Side notes:
* The order is important. We cannot say 64:1 because the old volume is mentioned first
* We can write the equation y = 64*x where x is the old volume and y is the new volume.