Question

Why is a number raised to a negative power the reciprocal of that number?

Answer 1

Simple answer:

We'll do this by working backwards.

How can you make `2^2` out of `2^3`?
Well, you divide by 2: `2^3/2 = 2^2`

How can you make `2^1` out of `2^2`?
Well, you divide by 2: `2^2/2 = 2^1`

How can you make `2^0 (=1)` out of `2^1`?
Well, you divide by 2: `2^1/2 = 2^0 = 1`

How can you make `2^-1` out of `2^0`?
Well, you divide by 2: `2^0/2 = 2^-1 = 1/2`

Proof why this should be the case

The definition of the reciprocal is: "a number's reciprocal multiplied by that number should give you 1".

Let `a^x` be the number.
`a^x * 1/a^x = 1`
Or you can also say the following:
`a^x*a^-x = a^(x+(-x)) = a^(x-x) = a^0 = 1`

Since both of these are equal to `1`, you can set them equal:
`a^x*a^-x = a^x*1/a^x`
Divide both sides by `a^x`:
`a^-x = 1/a^x`

And you have your proof.