Editing Exponentiation (section)

==Irrationality and transcendence==
{{Main|Gelfond–Schneider theorem}}
If {{mvar|b}} is a positive real [[algebraic number]], and {{mvar|x}} is a rational number, then {{math|''b''<sup>''x''</sup>}} is an algebraic number. This results from the theory of [[algebraic extension]]s. This remains true if {{mvar|b}} is any algebraic number, in which case, all values of {{math|''b''<sup>''x''</sup>}} (as a [[multivalued function]]) are algebraic. If {{mvar|x}} is [[irrational number|irrational]] (that is, ''not rational''), and both {{mvar|b}} and {{mvar|x}} are algebraic, Gelfond–Schneider theorem asserts that all values of {{math|''b''<sup>''x''</sup>}} are [[transcendental number|transcendental]] (that is, not algebraic), except if {{mvar|b}} equals {{math|0}} or {{math|1}}.

In other words, if {{mvar|x}} is irrational and <math>b\not\in \{0,1\},</math> then at least one of {{mvar|b}}, {{mvar|x}} and {{math|''b''<sup>''x''</sup>}} is transcendental.