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I was wondering, why do we use a radical sign $\sqrt{}$ for roots? Why don't we use subscripts instead? It would make sense too, regarding the fact that $\sqrt{5^3} = 5^{3/2}$. A switch like that would look like $$ 5^3_2 = 5^{\frac{3}{2}} $$

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    $\begingroup$ Subscripts are used for other things, such as distinguishing variables $x_1, x_2, \ldots$. $\endgroup$
    – vadim123
    Commented Nov 7, 2016 at 3:45
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    $\begingroup$ why don't we use superscripts, instead of subscripts? Like $5^{\frac23}$. Well, I see $5^3_2$, you could use it ... but it may take years for others to follow :) $\endgroup$
    – Mirko
    Commented Nov 7, 2016 at 3:55
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    $\begingroup$ There is no specific reason. It is the end result of a process of natural selection among mathematicians/scientists. $\endgroup$ Commented Nov 7, 2016 at 3:58

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It's arbitrary $-$ and that's okay. Many things in mathematics and science are arbitrary. It is arbitrary, for instance, that in physics up and right are considered positive directions, and down and left negative directions.

We say that the ratio of a circle's circumference to its diameter is $\pi$. Sure, the word pi comes from the Greek word for perimeter, and its symbol is the 16th letter in the Greek alphabet, but is there really any good reason for why we spelled it that way or picked the 16th letter? Not really.

The very nature of language itself is arbitrary: when we say "apple", there is nothing inherent in that sequence of sounds that must mean the fruit we think of. We English-speakers as a culture have simply agreed that it does. Language is nothing more than a cultural convention to arbitrarily encode meaning in sound (or shapes for written language, components in sign language).

Similarly, we could have very well used subscripts for roots. It was established, however, that a radical sign would be used, and since then we, the mathematical community, have agreed that

$$\sqrt x $$ denotes the square root of x.

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    $\begingroup$ I would add though, that a new symbol or a notation (or a word) has to be coherent with the language it belongs to. By this, I mean consistent, but also convenient. Imagine that the notation to putting $x$ to the $n$-th root was circling it $n$ times... That wouldn't even make sense! The radical notation has proven itself handy, even when used with different powers, so it stayed through time. $\endgroup$
    – Right Leg
    Commented Nov 7, 2016 at 4:39
  • $\begingroup$ i definitely agree - good point! $\endgroup$
    – Horse
    Commented Nov 7, 2016 at 5:17

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