Why Does $a^0 = 1$ and $a^{-p} = {\frac{1}{a^p}}$ if $(a \not = 0) , p \not= 0$?

Question

Why does $a^0 = 1$ and $a^{-p} = {\frac{1}{a^p}}$ if $(a \not = 0)$ and $p \not= 0$?

How can we prove these formulas?

Answer 1

This is just an intuition test of why the properties should be defined like this:

The fundamental property of the exponential functions is the following: $b^x\cdot b^y=b^{x+y}$.

The other properties are derived in such a way that the mentioned property is satisfied. For example: since we want the equation to be fulfilled $$b^0\cdot b^x=b^{0+x}=b^x$$ we must define $b^0:=1$. Since we want the following equation to be fulfilled $$b^{-x}\cdot b^x=b^{-x+x}=b^0=1$$ we must define $b^{-x}:=1/b^x$. Since we want the following equation to be fulfilled $$\underbrace{b^{1/n}\cdot b^{1/n}\cdots b^{1/n}}_{n\, \textrm{factors}}=b^1=b$$ we must define $b^{1/n}:=\sqrt[n]{b}$. Similarly, we must to define $b^{m/n}:=(\sqrt[n]{b})^m.$

Answer 2

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

Also,

Do you mean

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