Why does $a^{b}·a^{c}=a^{b+c}$? Why does $a^{b}·a^{c}=a^{b+c}$ ?

I want proof for it, I asked professor and he replied: "I don't know, it's a property anyway that is true and that's all what you need to  know."
It is easy to see this using the definition of a power , if $b$ and $c$ are natural numbers.

$$a^b \cdot a ^c = \overbrace{a \cdots a}^{b\ times} \cdot \overbrace{a \cdots a}^{c\ times} = \overbrace{a \cdots a}^{b+c\ times}
=a^{b+c}$$

With the rules

$$a^{-b}=\frac{1}{a^b}$$

and

$$a^{\frac{1}{b}}=\sqrt[b]{a}$$

it should be possible to extend it to the negative and rational numbers.
HaHa, interesting! 

Let us define an operation "$*$" on the set of positive integers $\mathbb{Z}_+\times \mathbb{Z}_+\to \mathbb{Z}_+$ as follows:
$a*b=\underbrace{b\times b\times\cdots \times b}_{a\text{
    times}}=b^a(\text{ usual exponentiation here })$. Define $a^2:=
a*a$, inductively define
$a^{n+1}:=a^{n}*a$. Then $a^3=a^2*a\neq a*a^2$ in general. For
example, 
$3^2*3=\underbrace{3\times \cdots \times 3}_{27\text{ times}}$
but $3*3^2=27\times 27\times 27$. Therefore $3*3^2\neq 3^3$.

Just for fun!