Definition:Power (Algebra)

Integers
Let $$x \in \mathbb{R}$$ be a real number.

Let $$n \in \mathbb{Z}$$ be an integer

The expression $$x^n$$ is called:
 * $$x$$ to the power of $$n$$, or
 * the $$n$$th power of $$x$$, or
 * $$x$$ to the $$n$$th power, or
 * $$x$$ to the $$n$$th, or
 * $$x$$ to the $$n$$

and is defined recursively as:

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

This agrees with the definition as given in Powers of Group Elements, which is appropriate as, under multiplication, the real numbers (less zero) form a group.

See below for the definition of $$x^n$$ where $$x = 0$$.

Even Power
Let $$x \in \mathbb{R}$$ be a real number.

Let $$n \in \mathbb{Z}$$ be an even integer.

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

Odd Power
Let $$x \in \mathbb{R}$$ be a real number.

Let $$n \in \mathbb{Z}$$ be an odd integer.

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

Rational Numbers
Let $$x \in \mathbb{R}$$ be a real number such that $$x > 0$$.

Let $$m \in \mathbb{Z}$$ be an integer.

Let $$y = \sqrt [m] x$$ be the $m$th root of $x$.

Then we can write $$y = x^{1/m}$$ which means the same thing as $$y = \sqrt [m] x$$.

Thus we can define the power of a rational number:

Let $$r = \frac p q \in \mathbb{Q}$$ be a positive rational number where $$p \in \mathbb{Z}, q \in \mathbb{Z} - \left\{{0}\right\}$$.

Then $$x^r$$ is defined as:

$$x^r = x^{p/q} = \left({\sqrt [q] x}\right)^p = \sqrt [q] {\left({x^p}\right)}$$.

When $$r = \frac {-p} q \in \mathbb{Q}: r < 0$$ we define:

$$x^r = x^{-p/q} = \frac 1 {x^{p/q}}$$ analogously for the negative integer definition.

See below for the definition of $$x^r$$ where $$x = 0$$.

Power of Zero
Let $$z \in \mathbb{R}$$ be a real number.

(This includes the situation where $$x \in \mathbb{Z}$$ or $$x \in \mathbb{Q}$$.)

When $$x=0$$, $$x^z$$ is defined as follows:

$$0^z = \begin{cases} 1 & : z = 0 \\ 0 & : z > 0 \\ \mbox{Undefined} & : z < 0 \\ \end{cases}$$

This takes account of the awkward case $$0^0$$: it is "generally accepted" that $$0^0 = 1$$ as this convention agrees with certain general results which would otherwise need a special case.