The universe is often described as being 4-dimensional (having 3 spatial dimensions + the dimension of time).
Some theories (such as some string theories) consider the universe as having even more dimensions. In some cases, some of the dimensions are viewed as "curled up".
For example: according to many versions of string theory, the universe could have 10 dimensions. Assuming that these theories are supported by relevant mathematics, it should be possible to explain the conclusion of the amount of dimensions mathematically. The equations used for this explanation therefore would use some notation containing the amount of dimensions.
What notation is used for the dimensions in physics equations? Are they indicated as exponents, and if so, are there cases when they are not present as exponents?