Editing Exponentiation (section)

===Particular bases===

====Powers of ten <span class="anchor" id="Base 10"></span>====
{{See also|Scientific notation}}
{{Main|Power of 10}}
In the base ten ([[decimal]]) number system, integer powers of {{math|10}} are written as the digit {{math|1}} followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, {{math|1={{val|e=3}} = {{val|1000}}}} and {{math|1={{val|e=-4}} = {{val|0.0001}}}}.

Exponentiation with base {{math|[[10 (number)|10]]}} is used in [[scientific notation]] to denote large or small numbers. For instance, {{val|299792458|u=m/s}} (the [[speed of light]] in vacuum, in [[metres per second]]) can be written as {{val|2.99792458|e=8|u=m/s}} and then [[approximation|approximated]] as {{val|2.998|e=8|u=m/s}}.

[[SI prefix]]es based on powers of {{math|10}} are also used to describe small or large quantities. For example, the prefix [[Kilo-|kilo]] means {{math|1={{val|e=3}} = {{val|1000}}}}, so a kilometre is {{val|1000|u=metres}}.

===={{anchor|Base 2}}Powers of two====
{{Main|Power of two}}
The first negative powers of {{math|2}} have special names: <math>2^{-1}</math>is a ''[[one half|half]]''; <math>2^{-2}</math> is a ''[[4 (number)|quarter]].''

Powers of {{math|2}} appear in [[set theory]], since a set with {{math|''n''}} members has a [[power set]], the set of all of its [[subset]]s, which has {{math|2<sup>''n''</sup>}} members.

Integer powers of {{math|2}} are important in [[computer science]]. The positive integer powers {{math|2<sup>''n''</sup>}} give the number of possible values for an {{math|''n''}}-[[bit]] integer [[binary number]]; for example, a [[byte]] may take {{math|1=2<sup>8</sup> = 256}} different values. The [[binary number system]] expresses any number as a sum of powers of {{math|2}}, and denotes it as a sequence of {{math|0}} and {{math|1}}, separated by a [[binary point]], where {{math|1}} indicates a power of {{math|2}} that appears in the sum; the exponent is determined by the place of this {{math|1}}: the nonnegative exponents are the rank of the {{math|1}} on the left of the point (starting from {{math|0}}), and the negative exponents are determined by the rank on the right of the point.

====Powers of one====
Every power of one equals: {{math|1=1<sup>''n''</sup> = 1}}.

====Powers of zero====
For a positive exponent {{math|''n'' > 0}}, the {{mvar|n}}th power of zero is zero: {{math|1=0<sup>''n''</sup> = 0}}. For a negative exponent, <math>0^{-n}=1/0^n=1/0</math> is undefined.

In some contexts (e.g., [[combinatorics]]), the expression [[zero to the power of zero|{{math|0<sup>0</sup>}}]] is defined to be equal to <math>1</math>; in others (e.g., [[Mathematical analysis|analysis]]), it is often undefined.

====Powers of negative one====
Since a negative number times another negative is positive, we have:<blockquote><math>(-1)^n = \left\{\begin{array}{rl} 
1 & \text{for even } n, \\
-1 & \text{for odd } n. \\
\end{array}\right.</math></blockquote>Because of this, powers of {{math|−1}} are useful for expressing alternating [[sequence]]s. For a similar discussion of powers of the complex number {{math|''i''}}, see ''{{slink||nth roots of a complex number}}''.