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Is this Notation correct?

For example:

$$\sqrt[-100]{100}$$

I think this is wrong, because

$$100^{\frac{1}{-100}}=100^{\frac{-1}{100}}=\sqrt[100]{100^{-1}}=\sqrt[100]{\frac1{100}}$$

Am I correct?

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    $\begingroup$ I think this notation is correct but should not be used to avoid confusion. $\endgroup$ – Mohammad Zuhair Khan Dec 27 '17 at 11:49
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The notation $\sqrt[-100]{100}$ is correct, albeit not commonly used. In fact the whole equality chain \begin{align*} \color{blue}{\sqrt[-100]{100}}=100^{\frac{1}{-100}}=100^{\frac{-1}{100}}=\sqrt[100]{100^{-1}}=\sqrt[100]{\frac1{100}}\tag{1} \end{align*} is correct.

Sometimes we can read in analysis books a definition of rational powers which goes like: Let $a>0$ and $r=\frac{p}{q}$ where $p,q$ are integers, $q>0$, then we define \begin{align*} a^r\equiv a^{\frac{p}{q}}:=\sqrt[q]{a^p}\tag{2} \end{align*}

Note that the definition above justifies only the following representations in (1) \begin{align*} 100^{\frac{-1}{100}}=\sqrt[100]{100^{-1}}=\sqrt[100]{\frac1{100}} \end{align*}

But since we are allowed to use the notation \begin{align*} \frac{-p}{q}=-\frac{p}{q}=\frac{p}{-q} \end{align*} an extension of the notation (2) to each representation in (1) is admissible.

Note: As a plausibility check note that Wolfram Alpha accepts $\sqrt[-100]{100}$ and suggests the following simplified representation \begin{align*} \sqrt[-100]{100}=\frac{1}{\sqrt[50]{10}} \end{align*}

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it can be simplified to $$\frac{1}{(10^2)^{1/100}}=\frac{1}{10^{1/50}}$$

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    $\begingroup$ Sir, I don’t think the OP has invoked a square root notation. $\endgroup$ – user371838 Dec 27 '17 at 11:54
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    $\begingroup$ but i can see a square root notation above? or i will need glases $\endgroup$ – Dr. Sonnhard Graubner Dec 27 '17 at 11:55
  • $\begingroup$ I took it to be the $-100$th root. An extension of $\sqrt[3]{2}$, $\sqrt[4]{2}$ etc but with an unusual negative. $\endgroup$ – badjohn Dec 27 '17 at 12:02
  • $\begingroup$ It is in similar sense to $\sqrt[3] x, \sqrt[-5] x$, sir. $\endgroup$ – user371838 Dec 27 '17 at 12:04
  • $\begingroup$ i saw it also here themathpage.com/Alg/rational-exponents.htm $\endgroup$ – Dr. Sonnhard Graubner Dec 27 '17 at 12:09
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It's unusual and hence clarification would be advisable to avoid misinterpretation. With appropriate clarification then it would be acceptable. You can even invent your own notation provided that you define it. However, when clearer alternatives are easily available, what's the point?

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  • $\begingroup$ I saw this notation new, today $\endgroup$ – Newuser Dec 27 '17 at 11:53
  • $\begingroup$ Can you give us a link? It is possible that it may have made sense in the context. $\endgroup$ – badjohn Dec 27 '17 at 11:58

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