Editing Exponentiation (section)

==Rational exponents==
[[File:Mplwp roots 01.svg|thumb|From top to bottom: {{math|''x''<sup>1/8</sup>}}, {{math|''x''<sup>1/4</sup>}}, {{math|''x''<sup>1/2</sup>}}, {{math|''x''<sup>1</sup>}}, {{math|''x''<sup>2</sup>}}, {{math|''x''<sup>4</sup>}}, {{math|''x''<sup>8</sup>}}.]]

If {{mvar|x}} is a nonnegative [[real number]], and {{mvar|n}} is a positive integer, <math>x^{1/n}</math> or <math>\sqrt[n]x</math> denotes the unique nonnegative real [[nth root|{{mvar|n}}th root]] of {{mvar|x}}, that is, the unique nonnegative real number {{mvar|y}} such that <math>y^n=x.</math>

If {{mvar|x}} is a positive real number, and <math>\frac pq</math> is a [[rational number]], with {{mvar|p}} and {{mvar|q > 0}} integers, then <math display="inline">x^{p/q}</math> is defined as
:<math>x^\frac pq= \left(x^p\right)^\frac 1q=(x^\frac 1q)^p.</math>
The equality on the right may be derived by setting <math>y=x^\frac 1q,</math> and writing <math>(x^\frac 1q)^p=y^p=\left((y^p)^q\right)^\frac 1q=\left((y^q)^p\right)^\frac 1q=(x^p)^\frac 1q.</math>

If {{mvar|r}} is a positive rational number, {{math|1=0<sup>''r''</sup> = 0}}, by definition.

All these definitions are required for extending the identity <math>(x^r)^s = x^{rs}</math> to rational exponents.

On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real {{mvar|n}}th root, which is negative, if {{mvar|n}} is [[odd number|odd]], and no real root if {{mvar|n}} is even. In the latter case, whichever complex {{mvar|n}}th root one chooses for <math>x^\frac 1n,</math> the identity <math>(x^a)^b=x^{ab}</math> cannot be satisfied. For example,
:<math>\left((-1)^2\right)^\frac 12 = 1^\frac 12= 1\neq (-1)^{2\cdot\frac 12} =(-1)^1=-1.</math>

See ''{{slink||Real exponents}}'' and ''{{slink||Non-integer powers of complex numbers}}'' for details on the way these problems may be handled.