Editing Exponentiation (section)

{{Short description|Arithmetic operation}}
{{Redirect|Exponent}}
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[[File:Expo02.svg|thumb|Graphs of {{math|1=''y'' = ''b''<sup>''x''</sup>}} for various bases {{math|''b''}}:
{{nobr|{{legend-line|inline=yes|green solid 2px|[[#Powers of ten|base&nbsp;{{math|10}}]],}}}}
{{nobr|{{legend-line|inline=yes|red solid 2px|[[Exponential function|base&nbsp;{{math|''e''}}]],}}}}
{{nobr|{{legend-line|inline=yes|blue solid 2px|[[#Powers of two|base&nbsp;{{math|2}}]],}}}}
{{nobr|{{legend-line|inline=yes|cyan solid 2px|base&nbsp;{{math|{{sfrac|1|2}}}}.}}}}
Each curve passes through the point {{math|(0, 1)}} because any nonzero number raised to the power of {{math|0}} is {{math|1}}. At {{math|1=''x'' = 1}}, the value of {{math|''y''}} equals the base because any number raised to the power of {{math|1}} is the number itself.]]
{{Arithmetic operations}}

In [[mathematics]], '''exponentiation''', denoted {{math|''b''<sup>''n''</sup>}}, is an [[operation (mathematics)|operation]] involving two numbers: the ''base'', {{mvar|b}}, and the ''exponent'' or ''power'', {{mvar|n}}.<ref name=":1">{{cite web |last=Nykamp |first=Duane |title=Basic rules for exponentiation |website=Math Insight |url=https://mathinsight.org/exponentiation_basic_rules |access-date=August 27, 2020}}</ref> When {{mvar|n}} is a positive [[integer]], exponentiation corresponds to repeated [[multiplication]] of the base: that is, {{math|''b''<sup>''n''</sup>}} is the [[product (mathematics)|product]] of multiplying {{mvar|n}} bases:<ref name=":1"/>
<math display="block">b^n = \underbrace{b \times b \times \dots \times b \times b}_{n \text{ times}}.</math>In particular, <math>b^1=b</math>.

The exponent is usually shown as a [[superscript]] to the right of the base as {{math|''b''<sup>''n''</sup>}} or in computer code as <code>b^n</code>. This [[binary operation]] is often read as "{{mvar|b}} to the power {{mvar|n}}"; it may also be referred to as "{{mvar|b}} raised to the {{mvar|n}}th power", "the {{mvar|n}}th power of {{mvar|b}}",<ref>{{MathWorld |title=Power |id=Power |access-date=2020-08-27}}</ref> or, most briefly, "{{mvar|b}} to the {{mvar|n}}".

The above definition of <math>b^n</math> immediately implies several properties, in particular the multiplication rule:<ref group="nb">There are three common notations for [[multiplication]]: <math>x\times y</math> is most commonly used for explicit numbers and at a very elementary level; <math>xy</math> is most common when [[variable (mathematics)|variable]]s are used; <math>x\cdot y</math> is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.</ref>

<math display="block">\begin{align}
b^n \times b^m & = \underbrace{b \times \dots \times b}_{n \text{ times}} \times \underbrace{b \times \dots \times b}_{m \text{ times}}  \\[1ex]
&  = \underbrace{b \times \dots \times b}_{n+m \text{ times}} \ =\ b^{n+m} . 
\end{align}</math>

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add. Extending this rule to the power zero gives <math>b^0 \times b^n = b^{0+n} = b^n</math>, and, where {{mvar|b}} is non-zero, dividing both sides by <math>b^n</math> gives <math>b^0 = b^n / b^n = 1</math>. That is the multiplication rule implies the definition <math display="block">
b^0=1.
</math>A similar argument implies the definition for negative integer powers: <math display="block">b^{-n} = 1/b^n.</math>That is, extending the multiplication rule gives <math>b^{-n} \times b^n = b^{-n+n} = b^0 = 1 </math>. Dividing both sides by <math>b^n</math> gives <math>b^{-n} = 1 / b^n</math>. This also implies the definition for fractional powers: <math display="block">b^{n/m} = \sqrt[m]{b^n}.</math>For example, <math> b^{1/2} \times b^{1/2} = b^{1/2 \,+\, 1/2} = b^1 = b  </math>, meaning <math> (b^{1/2})^2 = b </math>, which is the definition of square root: <math>b^{1/2} = \sqrt{b} </math>.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define <math>b^x</math> for any positive real base <math>b</math> and any real number exponent <math>x</math>. More involved definitions allow [[complex numbers|complex]] base and exponent, as well as certain types of [[matrix (mathematics)|matrices]] as base or exponent.

Exponentiation is used extensively in many fields, including [[economics]], [[biology]], [[chemistry]], [[physics]], and [[computer science]], with applications such as [[compound interest]], [[population growth]], [[chemical reaction kinetics]], [[wave]] behavior, and [[public-key cryptography]].