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:

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

This agrees with the definition as given in Power of an Element, 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 $r = \dfrac p q \in \Q$ 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 $r = \dfrac {-p} q \in \Q: r < 0$ we define:

$x^r = x^{-p/q} = \dfrac 1 {x^{p/q}}$ 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:

$x^r := \exp \left({r \ln x}\right)$

where $\exp$ denotes the exponential function.

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

This follows from Logarithms of Powers and Exponential of Natural Logarithm: 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

$z^k := e^{k \operatorname{Log} \left({z}\right)}$

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 $r \in \R$.

It is also called the index (plural indices).

Also see

• Root

Sources

• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.1$: Example $1$
• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.2$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.9$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 14.6$
• For a video presentation of the contents of this page, visit the Khan Academy.