Exponentiation

Definition: $a^x$ is defined as:

1) For $x\;\in\mathbb{Z^+}$:

$\qquad\quad a^x = \underbrace{(a.a.a.....a)}_\text{$x \text{ times}$}\\[6pt]
\qquad\quad a^{-x} = \dfrac{1}{a^x}$

$\;\;\;$For $a>0,\\[6pt]
\text{If}\;\;\quad\quad y^{x} = a\\
\text{Then,} \quad a^{1/x} = y$

2) For $x\;\in\mathbb{Q}$, then $x = p/q$, where, $p, q\;\in\mathbb{Z^+}$

$\;\;\;$For $a>0,\\[6pt]
\text{If}\;\;\quad\quad\;\; a^p = b^q\\[3pt]
\text{Then,} \quad a^{\tfrac{p}{q}} = b$

3) For $x\;\in\mathbb{R}$ and $a>0$:
$\;\;\; a^x$ for irrational numbers is defined in a way so as to have the continuous exponential function.