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Recently I became curious about when the following ideas came about, and I couldn't really find information about them with some google searches.

$a^0 = 1$

$a^{\frac pq} = \sqrt [q] {2^p}$

$a^{-x} = \dfrac {1}{a^x}$.

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This was essentially done by Napier, see Which came first, the natural logarithm or the base of the natural logarithm? and How did Napier come to invent logarithms? Napier was talking about the inverse function, but the tables he made can (and have to) be used both ways. The formal definition of the exponent for all numbers, even for complex ones is due to Euler, who used power series.

EDIT. Of course, as it always happens, Euler had predecessors. It is possible that the first work which discussed real (irrational) exponents is by Nicole Oresme, De proportionibus proportionum, written approximately in 1360. It seems that it has never been fully translated from the Latin, but there is a detailed exposition:

E. Grant, Nicole Oresme and His De Proportionibus Proportionum, Source: Isis, Vol. 51, No. 3 (Sep., 1960), pp. 293-314

The journal is available on JSTOR. Of course, Oresme did not have (and could not possibly have) a precise rigorous definition of irrational exponents, but nevertheless he discusses them, and some of his observations are very clever. In particular he understood that multiples of an irrational number are dense modulo 1, I think he was the first to state this clearly.

Remark. Of course, Euler perfectly understood real numbers, and exponents etc., but strictly speaking the formal definition of a real number was not available until the late 19th century.

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  • $\begingroup$ Are you sure that the notation of exponents was being used back then? From what I can find, it seems the work of Newton is where exponents beyond positive integers were first systematically used, particularly in his work on the binomial theorem, where the utility of the notation becomes quite vivid. See jstor.org/stable/2974078?seq=3#page_scan_tab_contents. $\endgroup$
    – KCd
    Commented Apr 15, 2017 at 21:09
  • $\begingroup$ I was answering the question: about the idea, not about notation. And Newton lived much later than Napier. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 16, 2017 at 20:33

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