Lets say the side of the cube is x units in length.
The volume of the cube then is x * x * x = x^3 units cubed
After the sides are doubled, new length of cubes is 2x units
New volume is (2x)^3 = 8(x)^3 units cubed.
The change in volume = 8(x)^3 divide by x^3 to get 8.
So the volume of the cube increases 8 times.