# Definition:Power (Algebra)

## Contents

- 1 Definition
- 2 Exponent
- 3 Examples
- 3.1 Sequence of Powers of 2
- 3.2 Sequence of Powers of 3
- 3.3 Sequence of Powers of 4
- 3.4 Sequence of Powers of 5
- 3.5 Sequence of Powers of 6
- 3.6 Sequence of Powers of 7
- 3.7 Sequence of Powers of 8
- 3.8 Sequence of Powers of 9
- 3.9 Sequence of Powers of 10
- 3.10 Sequence of Powers of 11
- 3.11 Sequence of Powers of 12

- 4 Also see
- 5 Sources

## 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:

- $e_m \left({n}\right) = \begin{cases} 1 & : n = 0 \\ m \times e_m \left({x}\right) & : n = x + 1 \end{cases}$

where:

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

$e_m \left({n}\right)$ is then expressed as a binary operation in the form:

- $m^n := e_m \left({n}\right)$

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 quotient.

### 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:

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

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.

### 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 any 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 and $\ln$ is the natural logarithm multifunction.

### Multiindices

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

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

Then $x^k$ is defined as:

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

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

## Examples

### Sequence of Powers of 2

The sequence of powers of $2$ begins:

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

This sequence is A000079 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 3

The sequence of powers of $3$ begins:

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

This sequence is A000244 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 4

The sequence of powers of $4$ begins:

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

This sequence is A000302 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 5

The sequence of powers of $5$ begins:

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

This sequence is A000351 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 6

The sequence of powers of $6$ begins:

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

This sequence is A000400 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 7

The sequence of powers of $7$ begins:

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

This sequence is A000420 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 8

The sequence of powers of $8$ begins:

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

This sequence is A001018 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 9

The sequence of powers of $9$ begins:

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

This sequence is A001019 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 10

The sequence of powers of $10$ begins:

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

This sequence is A011557 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 11

The sequence of powers of $11$ begins:

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

This sequence is A001020 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Sequence of Powers of 12

The sequence of powers of $12$ begins:

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

This sequence is A001021 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also see

- Definition:Root (Analysis)
- Definition:Exponential Function
- Exponentiation to Real Number is Extension of Exponentiation to Rational Number

## Sources

- For a video presentation of the contents of this page, visit the Khan Academy.

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary