Definition:Power (Algebra)

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Definition

Natural Numbers

Let $\N$ denote the natural numbers.


For each $m \in \N$, recursively define $e_m: \N \to \N$ to be the mapping:

$\map {e_m} n = \begin {cases} 1 & : n = 0 \\ m \times \map {e_m} x & : n = x + 1 \end{cases}$

where:

$+$ denotes natural number addition.
$\times$ denotes natural number multiplication.


$\map {e_m} n$ is then expressed as a binary operation in the form:

$m^n := \map {e_m} n$

and is called $m$ to the power of $n$.


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 $n$.

$x^n$ 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.


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 to a positive rational number:

Let $r = \dfrac p q \in \Q$ be a positive rational number where $p \in \Z_{\ge 0}, q \in \Z_{> 0}$.

Then $x^r$ is defined as:

$x^r = x^{p/q} = \paren {\sqrt [q] x}^p = \sqrt [q] {\paren {x^p} }$


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.


Real Numbers

We define $x^r$ as:

$x^r := \map \exp {r \ln x}$

where $\exp$ denotes the exponential function.


Complex Numbers

Let $z, k \in \C$ be complex numbers.


$z$ to the power of $k$ is defined as the multifunction:

$z^k := e^{k \ln \paren z}$

where:

$e^z$ is the exponential function
$\ln$ is the natural logarithm multifunction.


Multiindices

Let $k = \left \langle {k_j}\right \rangle_{j = 1, \ldots, n}$ be a multiindex indexed by $\set {1, \ldots, n}$.

Let $x = \tuple {x_1, \ldots, x_n} \in \R^n$ be an ordered tuple of real numbers.


Then $x^k$ is defined as:

$\ds x^k := \prod_{j \mathop = 1}^n x_j^{k_j}$

where the powers on the right hand side are integer powers.


Power of Zero

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

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

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

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

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 power operation $x^r$, the number $r$ is known as the exponent of $x$, particularly for $r \in \R$.


Also known as

The operation of raising a number to a power is also seen in some sources referred to as involution.

By a similar naming convention, extracting a root of a number is seen the same sources referred to as evolution.


Examples

Sequence of Powers of 2

The sequence of powers of $2$ begins:

$1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16 \, 384, 32 \, 768, \ldots$


Sequence of Powers of 3

The sequence of powers of $3$ begins:

$1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19 \, 683, 59 \, 049, \ldots$


Sequence of Powers of 4

The sequence of powers of $4$ begins:

$1, 4, 16, 64, 256, 1024, 4096, 16 \, 384, 65 \, 536, 262 \, 144, 1 \, 048 \, 576, 4 \, 194 \, 304, \ldots$


Sequence of Powers of 5

The sequence of powers of $5$ begins:

$1, 5, 25, 125, 625, 3125, 15 \, 625, 78 \, 125, 390 \, 625, 1 \, 953 \, 125, 9 \, 765 \, 625, \ldots$


Sequence of Powers of 6

The sequence of powers of $6$ begins:

$1, 6, 36, 216, 1296, 7776, 46 \, 656, 279 \, 936, 1 \, 679 \, 616, 10 \, 077 \, 696, \ldots$


Sequence of Powers of 7

The sequence of powers of $7$ begins:

$1, 7, 49, 343, 2401, 16 \, 807, 117 \, 649, 823 \, 543, 5 \, 764 \, 801, 40 \, 353 \, 607, \ldots$


Sequence of Powers of 8

The sequence of powers of $8$ begins:

$1, 8, 64, 512, 4096, 32 \, 768, 262 \, 144, 2 \, 097 \, 152, 16 \, 777 \, 216, 134 \, 217 \, 728, \ldots$


Sequence of Powers of 9

The sequence of powers of $9$ begins:

$1, 9, 81, 729, 6561, 59 \, 049, 531 \, 441, 4 \, 782 \, 969, 43 \, 046 \, 721, 387 \, 420 \, 489, \ldots$


Sequence of Powers of 10

The sequence of powers of $10$ begins:

$1, 10, 100, 1000, 10 \, 000, 100 \, 000, 1 \, 000 \, 000, 10 \, 000 \, 000, 100 \, 000 \, 000, 1 \, 000 \, 000 \, 000, \ldots$


Sequence of Powers of 11

The sequence of powers of $11$ begins:

$1, 11, 121, 1331, 14 \, 641, 161 \, 051, 1 \, 771 \, 561, 19 \, 487 \, 171, 214 \, 358 \, 881, 2 \, 357 \, 947 \, 691, \ldots$


Sequence of Powers of 12

The sequence of powers of $12$ begins:

$1, 12, 144, 1728, 20 \, 736, 248 \, 832, 2 \, 985 \, 984, 35 \, 831 \, 808, 429 \, 981 \, 696, \ldots$


Also see

  • Results about powers can be found here.


Sources

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