# Definition:Power (Algebra)/Integer

## Definition

Let $x \in \R$ be a real number.

Let $n \in \Z$ be an integer.

The expression $x^n$ is called $x$ to the power of $n$.

$x^n$ is defined recursively as:

$x^n = \begin{cases} 1 & : n = 0 \\ & \\ x \times x^{n - 1} & : n > 0 \\ & \\ \dfrac {x^{n + 1} } x & : n < 0 \end{cases}$

where $\dfrac{x^{n + 1} } x$ denotes quotient.

### Even Power

Let $x \in \R$ be a real number.

Let $n \in \Z$ be an even integer.

Then $x^n$ is called an even power of $x$.

### Odd Power

Let $x \in \R$ be a real number.

Let $n \in \Z$ be an odd integer.

Then $x^n$ is called an odd power of $x$

## Also known as

The expression $x^n$ is vocalised in a number of other ways:

• the $n$th power of $x$
• $x$ to the $n$th power
• $x$ to the $n$th
• $x$ to the $n$.

### Knuth Uparrow Notation

In certain contexts in number theory, the symbol $\uparrow$ is used to denote the (usually) integer power operation:

$x \uparrow y := x^y$

This notation is usually referred to as Knuth (uparrow) notation.

## Examples

### Negative Power: $-3^{-3}$

$-3^{-3} = -\dfrac 1 {27}$

## Historical Note

The concept of an integer power to a negative exponent was introduced by John Wallis in the $17$th century.