Is this Notation correct : $\sqrt[-100]{100}?$

Question

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?

Answer 1

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*}

Answer 2

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

Answer 3

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?