Why Does n^0=1? Understanding the Exponent Rule in Math

[Universe_Man]

I always learned this in math, but never really questioned it.

why does n^0=1

 

[LeonhardEuler]

I always learned this in math, but never really questioned it.

why does n^0=1

Alright, first you know x=x^1 and x^1x^1=x^{1+1}=x^2, right? And in general, x^ax^b=x^{a+b} when a and b are greater than or equal to one. So why not define x^0 so that this is true even if a or b is zero? If this is the case then
x^0x^a=x^{a+0}=x^a
We have x⁰x^a=x^a, so as long as x is not zero we can divide by x^a to come out with x^0=1.

More generally, powers are defined so that x^ax^b=x^{a+b} is true even when a and b are not integers and even when they are not real.

 

[Markjdb]

A simple proof:

Because (n^a)*(n^b) = n^{a+b},
it can be said that n^0 is equivalent to n^1 * n^{-1}.
n * (1/n) = 1,
therefore, n^0 = 1

 

[Universe_Man]

OH OH ok I get it, thanks a lot guys.

 

[HallsofIvy]

In other words, if we want a^maⁿ= a^n+m to be true for 0 as well as positive integer value of m and n, we must define a⁰= 1.
It is also true that if we want a^maⁿ= a^n+m to be true for negative integer powers, then we must have aⁿa^-n= a⁰= 1. In other words, we must define a^{-n}= \frac{1}{n}.

To see how we must define a^{\frac{1}{n}}, look at the law
(aⁿ)^m= a^mn.