# Exponentiation

## Concept of Exponentiation

Exponentiation refers to the mathematical operation an, where the number n is called the exponent and the number a is called the base of the exponentiation. The inverse operations of exponentiation are taking roots and logarithms.

If n is a natural number, then:

\begin{align} a^{n}&=\underset{n}{\underbrace{a\times a \times ...\times a}}\\ \\ \textrm{if}\; n&\in \mathbb{N_{1}}\\ \\ \mathbb{N_{1}} &=\left\{1;2;3; ...\right\}\\ \end{align}

\begin{align} a^{1}&=a\\ \\ a^{0}&=1,\; \textrm{if}\; a\neq 0 \\ \end{align}

For a negative exponent, exponentiation involves division:

\begin{align} a^{-n}&=\frac{1}{a^{n}},\\ \\ \textrm{if}\; a&\neq 0\; \textrm{and}\; n\in \mathbb{Z}\; \textrm{or}\\ \textrm{if}\; a&> 0\; \textrm{and}\; n\in \mathbb{Q}\\ \\ \mathbb{Z} &=\left\{\pm 1; \pm 2; \pm 3; ...\right\} \\ \mathbb{Q} &=\left\{\frac{b}{c},\; \textrm{where}\; b,c\in \mathbb{Z}, d\neq 0 \right\} \end{align}

For a rational (fractional) exponent, exponentiation involves taking roots:

\begin{align} a^{\frac{m}{n}}&=\sqrt[n]{a^{m}},\\ \\ \textrm{if}\; a&> 0, m\in \mathbb{Z}\; \textrm{and}\; n\in \mathbb{N_{2}}\\ \\ \mathbb{N_{2}} &=\left\{2;3;4; ...\right\}\\ \end{align}

## Properties of Exponents

\begin{align} a&> 0 \Rightarrow a^{n} > 0,\; \textrm{if}\; n\in \mathbb{R}\\ \\ (-a)^{2n}&=a^{2n}\\ \\ (-a)^{2n+1}&=-a^{2n+1}\\ \\ 0^{n}&=0,\; \textrm{if}\; n > 0\\ \\ 1^{n}&=1\\ \end{align}

Real numbers (ℝ) include all rational and irrational numbers, i.e., all positive and negative numbers, zero, and both algebraic and transcendental numbers.

## Operations with Exponents

\begin{align} a^{m}\times a^{n}&= a^{m+n}\\ \\ \frac {a^{m}}{a^{n}}&= a^{m-n}\\ \\ (a^{m})^{n} &= a^{m\times n}\\ \\ (a^{m})^{\frac{1}{n}} &= a^{\frac{m}{n}}\\ \\ a^{n}\times b^{n}&= (a \times b)^{n}\\ \\ \frac {a^{n}}{b^{n}}&= \left (\frac {a}{b} \right )^{n}\\ \\ \left (\frac {a}{b} \right )^{-n}&= \left (\frac {b}{a} \right )^{n},\; \textrm{where}\; a,b\neq 0\\ \\ \frac {1}{a^{-n}}&= a^{n}\\ \\ \frac {1}{a^{n}}&= a^{-n}\\ \end{align}