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 division.

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 $n$.

Also see

 * Definition:Power of Zero for the definition of $x^n$ where $x = 0$.


 * Non-Zero Real Numbers under Multiplication form Abelian Group


 * Definition:Power of Group Element, where the operation is defined in a general group and shown to be consistent with the definition given here.