Editing Surreal number (section)

===Other exponentials===
The [[#Powers of ω|powers of {{mvar|ω}}]] function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-{{mvar|e}} exponential, and it is this function that is meant whenever the notation {{mvar|ω{{sup|x}}}} is used in the following.

When {{mvar|y}} is a dyadic fraction, the [[power function]] <math display=inline>x \in \mathbb{No}</math>, {{math|''x'' ↦ ''x''{{sup|''y''}}}} may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation {{math|1=''x''{{sup|''y''+''z''}} = ''x{{sup|y}}'' · ''x{{sup|z}}''}}, and where defined it necessarily agrees with any other [[exponentiation]] that can exist.