# How to Compute Power Towers x^{xx ...}

A power tower (also known as a tetration or iterated exponential) is a mathematical operation that extends the idea of exponentiation. The n^{th} power of x, denoted by **x ^{n}**, means x multiplied by itself n times.

^{th}iterated exponent of x, usually denoted by

**x↑↑n**, and less often by

**,**

^{n}xmeans x exponentiated by itself n times. For instance, x↑↑2 = x

^{x}, and x↑↑3 = x

^{xx}, etc.

For numbers within a very small range, you can even define an infinite power tower (infinite tetration), in which you exponentiate a number by itself infinitely many times. You can calculate infinte tetration using the calculator on the left. For numbers outside of the given range, the operation does not converge to a single, finite number.

### Growth of Power Towers

Power Towers grow even more rapidly than exponentiated numbers. Consider the first, second, third, and fourth iterated exponents of 2:^{1}2 =

**2**

^{2}2 = 2

^{2}=

**4**

^{3}2 = 2

^{22}= 2

^{4}=

**16**

^{4}2 = 2

^{222}= 2

^{16}=

**65536**

But

^{5}2 = 2

^{65536}= is a number with 19729 digits. (You can see it here.)

### Infinite Power Towers

The infinite power tower x↑↑∞ (or^{∞}x) converges so long as

e

^{-e}< x < e

^{1/e}.

It can be written in closed form using the Lambert W function. The formula is

x

^{xxx ...}= -W[-ln(

*x*)]/ln(

*x*)

Some particular values of

*x*↑↑∞ are

1↑↑∞ = 1

√2↑↑∞ = 2

(1/e)↑↑∞ = 0.5671432904, the solution to

*x*= -ln(

*x*)

(1/4)↑↑∞ = 1/2

Zero is an example of a number whose power towers 0↑↑n do not converge as n goes to ∞. The value of 0↑↑n oscillates between 0 and 1 depending on whether n is even or odd

^{1}0 = 0↑↑1 =

**0**

^{2}0 = 0↑↑2 = 0

^{0}=

**1**

^{3}0 = 0↑↑3 = 0

^{00}= 0

^{1}=

**0**

^{4}0 = 0↑↑4 = 0

^{000}= 0

^{0}=

**1**

(Note that although 0

^{0}is an indeterminate form, in power towers it is defined as 1 since the limit of the function f(x) = x

^{x}goes to 1 as x goes to 0.)

© *Had2Know 2010
*