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Why do we write $a^n$ instead of $^n\!a$ for exponentiation? What benefit is there to writing the base before the exponent? With addition and multiplication order doesn't matter since $a + b = b + a$, so why was $a^n$ chosen, and who popularised this notation?

To me the first is preferable since it's a logical choice, as you put the number in front when you add: $$a+a+a = 3a$$ Thus it makes sense to write: $$a\times a\times a = \,^3\!a$$

It would also retain left-associativity as is commonly used with subtraction and division: $$(a\text{^} b)\text{^}c = \,^\left(^ab\right)c$$

Historically the best choice isn't always the one which got popularised, but there must have been some reason why this choice were made and preferred in its period of invention.

A similar question was previously asked on Mathematics.

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    $\begingroup$ Personally, I think that Descartes' choice for $a^3$ instead of $^3a$ was "more rational"; especially with "ancient" printing, it is quite easy to equivocate between $^3a$ and $3a$ ... $\endgroup$ – Mauro ALLEGRANZA Dec 21 '14 at 17:07
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    $\begingroup$ Note also that $^3a$ has some currency as tetration, i.e., $^3a=a^{a^a},$ providing also a modern reason to avoid that notation. (Personally, I dislike this notation, preferring Knuth's $\uparrow^2$, but that's neither here nor there.) $\endgroup$ – Charles Dec 21 '14 at 17:24
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    $\begingroup$ @fvel: That's why I called it a "modern reason". $\endgroup$ – Charles Dec 21 '14 at 17:28
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    $\begingroup$ @fvel I understand completely what you mean. I apologize if I came off as harsh; a couple weeks ago we had a cross-posting incident here from Mathematics where a question was cross-posted and an answer from Mathematics. There was another issue with a second answer (that ended up being deleted), a meta post, dialogue with a Math mod to solve it. . . The whole thing was a nightmare. I guess I was a little worried this might end up like that, so I sincerely apologize if I seemed hostile, rude or otherwise unpleasant. That certainly wasn't my intent! $\endgroup$ – HDE 226868 Dec 21 '14 at 22:03
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    $\begingroup$ @fvel note that merging of questions is an option we have here. $\endgroup$ – Danu Dec 22 '14 at 7:39
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Browsing the "original" historiographical" source for the "power" symbolisms , i.e. :

can be very istructive, showing how Descartes' choice was the last step of a complex process :

  • Pietro Antonio Cataldi : $5$ $3(crossed)$ for $5x^3$

  • Joost Bürgi : with $vi$ on top of $8$ for $8x^6$ (obvious problem : how to handle two variables : $x$ and $y$ ?)

  • Romanus (Adriaan van Roomen) : $A(4)$ for $A^4$

  • Pierre Hérigone : $a3$ for $a^3$ and $2b4$ for $2b^4$

  • James Hume (1636) : $A^{iii}$ for $A^3$.

Thus, Cajori concludes that :

Hérigone and Hume almost hit upon the scheme of Descartes. [...] Where Hume would write $5a^{iv}$ and Hérigone would write $5a4$, Descartes wrote $5a^4$.

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See Earliest Use of Symbols for Operations for a discussion. Summary: Descartes 1637 is the first to use exponents for powers in the modern way: $a^3$.

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    $\begingroup$ I think this answer could be significantly improved by expanding it a little bit; right now it's very short and not terribly enlightening. $\endgroup$ – Danu Dec 22 '14 at 7:36

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