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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.

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