# Notation for nested exponents

An expression such as $a^{b^c}$ is usually interpreted as $a^{(b^c)}$ and not as ${(a^b)}^c$. (See, for example, the Wikipedia entry for double exponential function.) Is there a reputable source that states how an expression such as $a^{b^{c^{\cdots^n}}}$ is to be interpreted?

## 1 answer

The power operation is considered right-associative, because the left-associative interpretation would just be a more complicated notation for the product of the exponents:

$a^{bc} = (a^b)^c \neq a^{b^c} = a^{(b^c)}$

Similar arguments lead to the same interpretation for higher towers, where it is even more apparent that this interpretation avoids redundancy.

$a^{bc...xy} = ((...(a^b)^c...)^x)^y\neq a^{b^{c^{⋰^{x^y}}}} = a^{(b^{(c^{⋰^{(x^y)}⋱})})}$

I could not quickly find a reference introducing the notation, the authors seem to use this notation without introducing it explicitly. However the following sources make the intended interpretation very clear with the corresponding definitions for Knuth's up-arrow notation [Wikipedia].

Knuth introduced the arrow notation and illustrates its use with powers of 10:

Similarly we can define a number I shall write as $x\uparrow n$ [...] For example $10\uparrow 10$ [...] is usually written as $10^{10}$

Knuth defines the $\uparrow\uparrow$ as $\uparrow$ nested from the right:

$x\uparrow\uparrow n = x\uparrow(x\uparrow(...\uparrow x)...))$

With the examples showing the correspondance between arrow and superscript notation it is clear that he also intends the same order of application for a power tower.

$10\uparrow\uparrow{\color{red}10} = 10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}$

where I inserted the the red ${\color{red}10}$ as I think it is a typo in the source.

See also Galidakis, Ioannis and Weisstein, Eric W. "Power Tower." From MathWorld--A Wolfram Web Resource.

Equation (1) links the repeated exponentiation notation to the double arrow notation. If we set $n=2$ in equations (2) to (4) we get

$\underbrace{a^{a^{⋰^{a}}}}_{k} $ $ = a\uparrow\uparrow k $ $ = a\uparrow[a\uparrow\uparrow (k-1)] $

$ = a^{\Big(\raise 4pt {[ {\underbrace{a^{a^{⋰^{a}}}}_{(k-1)}} ]}\Big)}$

Applying this definition recursively will nest the rightmost power into the innermost parenthesis.

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