There is an other way which is much more intuitive ,to prove the existence of extra dimensions beside the test in the last generation particle accelerators

The attractive force between two masses depend on :

distance^2

You can ask the questions :

- why the exponent is "2" instead of , for example , "1.99" or "2.11" ?
- why is it a constant ?
- why is it "2" also for the electric field ?

The answer is :

Because we live in a 3d world

The surface of a 3D sphere is proportional to : radius^2
If you put a mass in the center of the sphere, the intensity of the flux of gravitational field on each portion of the surface of the sphere depend on radius^2

The extra 7 dimensions are so small that , in general, they can not have any influence
However nowadays it is possible to measure the attractive force between two very close objects
The exponent "2" should increases if extra dimensions do really exist but all the tests proved that it remains constant

Cosequently the supporter of the string theories had to admit that the extra dimensions must be even much smaller than they expected

Obviusly it might be true but these tests cast a shadow on the theory