Definition:Power (Algebra)

Integers
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 $$n$$, 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 $$x \in \R$$ 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 $$x$$

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

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

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


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

When $$r = \frac {-p} q \in \Q: r < 0$$ 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 $$x = 0$$.

Real Numbers
Let $$x \in \R$$ be a real number such that $$x > 0$$.

Let $$r \in \R$$ be any real number.

Then we define $$x^r$$ as:


 * $$x^r \ \stackrel {\mathbf {def}} {=\!=} \ \exp \left({r \ln x}\right)$$

This definition is an extension of the definition for rational $r$.

This follows from Logarithms of Powers and Basic Properties of Exponential Function: it can be seen that $$\forall r \in \Q: \exp \left({r \ln x}\right) = \exp \left({\ln \left({x^r}\right)}\right) = x^r$$.

Complex Numbers
Let $$z, k \in \C \ $$ be any complex numbers. Then we define the power


 * $$z^k \ \stackrel {\mathbf {def}} {=\!=}$$ $$e^{k \operatorname{Log}(z)} \ $$

where $$e^x \ $$ is the exponential function and $$\operatorname{Log} \ $$ is the principal branch of the natural logarithm function.

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

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

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.

Exponent
In the expression $$x^r$$, the number $$r$$ is known as the exponent of $$x$$, particularly for $$r \in \R$$.