Negative powers

We know that aⁿ is for n = 0, 1, 2, 3, ...

We shall now find out what aⁿ is when n is negative.

We know that    and we used his to find a⁰ (we also implicitly used this to find a¹).

We shall now use this to find aⁿ when n is negative.

Similarly,

and

etc.

We could also have derived this in other ways.

One way is by saying that

and

     a^-n = (a^-1)ⁿ

Another way is by noting that

     aⁿ x a^-n = a^{n - n}

                  = a⁰

                  = 1

Thus a^-n x a^-n = 1

     So