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 xⁿ, means x multiplied by itself n times.

The n^th iterated exponent of x, usually denoted by

x↑↑n, and less often by ⁿx,

means 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 infinite 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:

¹2 = 2
²2 = 2² = 4
³2 = 2²² = 2⁴ = 16
⁴2 = 2²²² = 2¹⁶ = 65536

But ⁵2 = 2⁶⁵⁵³⁶ = 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

¹0 = 0↑↑1 = 0
²0 = 0↑↑2 = 0⁰ = 1
³0 = 0↑↑3 = 0⁰⁰ = 0¹ = 0
⁴0 = 0↑↑4 = 0⁰⁰⁰ = 0⁰ = 1

(Note that although 0⁰ 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