Question

How do you simplify `5^0`?

Answer 1

Please see below.

Explanation:

While it is easy to understand `5^2` and `5^5`, which are `5xx5` and `5xx5xx5xx5xx5`

it is not easy to understand `5^0`.

So let us try it different way.

What is `5^5-:5^2`?

To answer this let us use division as fraction `5^5/5^2` and this is

`(5xx5xx5xx5xx5)/(5xx5)`

= `(5xx5xx5xxcancel5xxcancel5)/(cancel5xxcancel5)`

= `5^3` and we can express this as `5^((5-2)` i.e.

`5^5-:5^2=5^((5-2)`

In fact similarly we can generalize it to say

`a^m-:a^n=a^m/a^n=a^((m-n)`, which is an identity

What is `a^0` then. Well we can use above identity and can interpret it as

`a^0=a^((m-m)`, but RHS is `a^m-:a^m` or `a^m/a^m` i.e. `1`

Hence for any `a`, we have `a^0=1` and so too for `a=5`

and `5^0=1`