Let the side of the cube be a,
Now If you double volume but keep it as a cube (with the new side length as b)
And so is your answer
A volume always involves (length) x (length) x (length), whether it's the volume of a cube, a cone, a sphere, or whatever. So similar solids (i.e. solids with the same proportions, but different measurements) will have volumes that vary according to the cube of their lengths.
For example, if we make a model that is one-sixth the size of the original, then the original's volume is 6 x 6 x 6 = 216 times as great as the model's.
In a similar way, the area of two similar shapes varies as the square of their lengths. So in the case of our model, the original's surface area will be 6 x 6 = 36 times that of the model.
When we work back from volume to length we must remember to use the inverse or 'opposite' operation to cubing: that is, finding the cube root. So, if we halve the volume - in other words we divide it by 2 - we will need to divide the lengths by the cube root of 2.
Similarly, working back from area to length will involve a square root. So, if an area is halved, then lengths will be divided by the square root of 2.
I hope that this explanation may supplement ADARSH's reply and provide a 'feel' for what is happening here.