How to solve 1^n=1 for n=0?

Question

If it is not apparent from the question title, I am okay with math but not brilliant.

For the following:

$$ 1^n=1 $$

How is this solved for n=0 ?

Wolfram Alpha says this is true and explains any nonzero number to the zero power is one.

EDIT: To ask more clearly, how is it true that any nonzero number to the zero power is one when it seems from my limited understanding that one multiplied by itself zero times should be zero - I have zero ones in a row with the multiplication of each one.

I am asking for an elementary explanation.

Answer 1

By definition for $a\neq0$

$$a^n =\stackrel{\color{red}{\text{n terms}}} {a\cdot a \cdot a\cdot...\cdot a}$$

and

$$a^{n-m} =\frac{\stackrel{\color{red}{\text{n terms}}}{a\cdot a \cdot a\cdot...\cdot a}}{\stackrel{\color{red}{\text{m terms}}}{a\cdot a \cdot a\cdot...\cdot a}}$$

thus by definition

$$a^0=a^{n-n}=\frac{\stackrel{\color{red}{\text{n terms}}}{a\cdot a \cdot a\cdot...\cdot a}}{\stackrel{\color{red}{\text{n terms}}}{a\cdot a \cdot a\cdot...\cdot a}}=1$$