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

Dec 18, 2014

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 ${2}^{0} \left(= 1\right)$ 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: ${2}^{0} / 2 = {2}^{-} 1 = \frac{1}{2}$

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.
${a}^{x} \cdot \frac{1}{a} ^ x = 1$
Or you can also say the following:
${a}^{x} \cdot {a}^{-} x = {a}^{x + \left(- x\right)} = {a}^{x - x} = {a}^{0} = 1$

Since both of these are equal to $1$, you can set them equal:
${a}^{x} \cdot {a}^{-} x = {a}^{x} \cdot \frac{1}{a} ^ x$
Divide both sides by ${a}^{x}$:
${a}^{-} x = \frac{1}{a} ^ x$