Question

Why are negative exponents used?

Answer 1

We use negative exponents because it's always more convenient to deal with functions (in this case exponential functions) defined on as wide set of arguments as possible.

It appears that exponential function can be defined not only for natural numbers, but also for rational, negative and even complex numbers, while preserving all the basic properties of this function defined for natural arguments.

In particular and most importantly, the obvious property of exponent `a^p*a^q=a^(p+q)`, which for natural `p` and `q` is a consequence of the definition, is completely preserved for rational, negative and even complex `p` and `q`. The proof in each case is relatively straight forward.

Another property of natural exponent is `(a^b)^c=a^(b*c)`.
This property is also preserved when we expand the definition of the exponent to rational, negative or complex numbers.

You can get more detailed information with corresponding proofs at Unizor by following menu items Algebra - Exponential Functions.