# My simple proof of x^0 = 1

Mute
Homework Helper
One can define exponentials in terms of the family of standard functions and it falls naturally that anything to the power zero is one. Exponentials are defined as follows:

$$n^x = e^{x\ln{n}}$$

Now put x = 0 and see what happens.
Even with technicalities about proving a^0 = 1 versus defining it, your argument there is circular. You want to "prove" that $a^0 = 1$, and then proceed to do so by writing it as $e^{x\ln a}$, and then set x = 0 and say, "Oh, e^0 = 1, so a^0 = 1", but you didn't prove that "e^0 = 1"! You used the result you were trying to prove in order to prove it!

Kurdt
Staff Emeritus
Gold Member
I didn't say I was trying to prove anything. Its merely the standard definition of an exponential function and shows how numbers to the power zero are defined to be 1. And yes I did not explain other definitions such as that of the exponential function, but I was taking it as a well established result.

Can you prove that $$\frac{a^n}{a^m} = a^{n-m}$$ when $$n = m$$? Usually the proof is only valid when they aren't equal and then people define $$x^0=1$$ such that the property remains valid for $$n = m$$.
perfect example of what will happen if people study more than actually required

arildno
Homework Helper
Gold Member
Dearly Missed
For a decreasing sequence of integers as exponents, we might use the following definition of exponentiation to prove the statement:
$$a^{n-1}=\frac{a^{n}}{a}, a\neq{0}, a^{1}=a, n\leq{1}$$

This definition is sufficient to prove that several properties we would like exponentiation to have actually hold.

a^0 = 1

FIRST: We use a one of the laws of exponents which is (a^m/a^n) = 1.

Let m=n,then (a^n/a^n) = a^(n-n) = a^0, where a is not equal to zero.

In short, (a^n/a^n) = a^0.

SECOND: There's a theorem that any nonzero number divided by itself is equal to 1.

That is, (a^n)/(a^n) = 1.

Combining the first and second by transitive property, we have a^0 = 1.

:)

$$1 = \frac{a^n}{a^n} = a^{n-n} = a^0$$

What could be more simpler than that?
I think this actually proves it the previous once doesn't really prove it, because you'd need the information above for it. Besides this is such a simple proof and anyone can understand it.