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square root with negative symbol on the left side of the tiny number 2

Does $\sqrt[-2]{Insert Any Number Here}$ exist?

what do you call it ?

negative square root?

also, where do negative square root ( $\sqrt[-2]{}$ ) mostly used or found? which mathematics topic you can find plenty of ( $\sqrt[-2]{}$ )?

and does ( $\sqrt[-2]{}$ ) appears frequently on calculus, abstract algebra, number theory , and combinatorics?

JustAsk123
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  • The answer below gives the result we get by simply using the definition , nevertheless such expressions are very unusual and probably almost never occur in papers or exercises. – Peter Jan 03 '24 at 06:34

1 Answers1

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If we define $\sqrt[a]{x}$ to be $x^{1/a}$, then naturally $\sqrt[-2]{x}$ is the same as $x^{1/(-2)}$, i.e. $x^{-1/2}$, i.e. $1/\sqrt{x}$.

So this does exist, though not necessarily "for any inserted number here," unless one is very careful for negative/complex numbers.

I don't think there's much of note to really discuss about this function. Does it appear in various branches of math? Will you find results stating it or utilizing it? Sure. But it's not like there's a deep connection nor is it some special function. It would be like saying $f(x)=x^2$ is something special; it pops up everywhere, but mostly as an artifact of its simplicity.

The closest it has to a "name" would be the "inverse square root", not that I've heard this used often in a mathematical context, so much as in programming (e.g. the infamous code segment from Quake III Arena (Wikipedia article)).

PrincessEev
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