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

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Answer 1 · 2014-12-18T20:50:15

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.