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I started wondering about this the other day. Since the following have their own alternate representations. $$\begin{align*} \displaystyle\large x+x=2x & \ \frac{x}{x}=1 & xx=x^2\end{align*}$$

Can $x^x$ be represented in some other way? Thanks.

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It could be written in many different ways algebraically. for example $x^x=\sqrt{x^{2x}}$ –  Emmad Kareem Oct 7 '12 at 1:17
$x/x=1$ is not true when $x=0$, careful. –  Pedro Tamaroff Oct 7 '12 at 1:17

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up vote 14 down vote accepted

Yes, it's called tetration. We can write $x^x$ as $^{2}x$.

There's actually a whole chain of these iterated operators, such as the (rather) famous Knuth up-arrow notation. The page I linked to has quite a few examples if you are interested.

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This is a nice way to represent it if you want to differentiate it, since you can then just apply the standard differentiation rules.

Something like $$x^{x^x}$$ will be represented as $$e^{e^{x\log(x)}\log(x)}$$

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