Question

What are Negative Exponents?

Answer 1

Negative exponents are an extension of the initial exponent concept.

To understand negative exponents ,
first review what we mean by positive (integer) exponents

What do we mean when we write something like:
`n^p` (for now, assume that `p` is a positive integer.

One definition would be that
`n^p` is `1` multiplied by `n`, `p` times.

Note that using this definition
`n^0` is `1` multiplied by `n`, `0` times
i.e. `n^0 = 1` (for any value of `n`)

Suppose you know the value of `n^p` for some particular values of `n` and `p`
but you would like to know the value of `n^q` for a value `q` less than `p`

For example suppose you knew that
`2^10 = 1024` but you wanted to know what `2^9` was equal to.
Is there a faster way than multiplying `1` by `2`, `9` times?
Yes.
If we note that `2^9 = (2^10)/2`
we can simply divide `1024` by `2` (giving 512) to obtain `2^9`

In general if we know that the value of `n^p` is `k`
and we want to know the value of `n^q` when `q<p`
we can simply divide k by n^(p-q)

With this in mind what is the value of
`n^(-t)` ?
We know that `n^0 = 1`
so `n^(-t)` must be `1` divided by `n`, `(0 - (-t))` times

That is `n^(-t) = 1/n^t`

As a final example consider the descending powers of 3 in the following, noting that with each line down the result is decreased by dividing the current value by 3

`3^4 = 81`
`3^3 = 27`
`3^2 = 9`
`3^1 = 3`
`3^0 = 1`
`3^(-1) = 1/3`
`3^(-2) = 1/9`
`3^(-3) = 1/27`