Mathematically, what is the correct way to write a speed of 100 meters per second?

Per means "divide". So meter per second means meter/second.

Also note that a negative power in the numerator is equivalent to a positive in the denominator.

Example:

1/s = s^-1

1 / (s^2) = s^-2

Based on this:

100 meters per seconds can be written as:

100 m/s, or

100 ms^-1