Power Tower — Definition, Formula & Examples

A power tower is a chain of exponents stacked on top of one another, such as

2^{3^{4}}

. You always evaluate a power tower from the top down (right to left), not from the bottom up.

A power tower (also called tetration when the base is repeated) is an expression of the form $a_1^{a_2^{a_3^{\cdots^{a_n}}}}$ , defined recursively by right-associative exponentiation: the topmost (rightmost) exponent is computed first, and each result becomes the exponent for the level below it.

Key Formula

a^{b^c} = a^{\left(b^c\right)}

Where:

$a$ = The base of the tower (bottommost number)
$b$ = The middle value
$c$ = The topmost exponent, evaluated first

How It Works

To evaluate a power tower, start at the very top of the stack and work downward. Compute the topmost exponentiation first, then use that result as the exponent for the next base below. For example, in

2^{3^2}

, you first compute

3^2 = 9

, then compute

2^9 = 512

. This right-to-left rule is critical — left-to-right evaluation gives a completely different (and incorrect) answer. Power towers grow extraordinarily fast; even modest-looking towers like

3^{3^3}

equal

3^{27} = 7{,}617{,}893{,}088

Worked Example

Problem: Evaluate the power tower

2^{3^2}

Step 1: Start at the top of the tower. Compute the topmost exponentiation.

3^2 = 9

Step 2: Use that result as the exponent for the base below it.

2^9 = 512

Answer:

2^{3^2} = 512

Why It Matters

Power towers appear in computer science when analyzing algorithms with hyper-exponential growth and in advanced number theory (for instance, bounds involving iterated exponentials). Understanding right-to-left evaluation also reinforces the conventions behind exponent notation that you rely on throughout algebra and calculus.

Common Mistakes

Mistake: Evaluating left-to-right (bottom-up), treating

2^{3^2}

(2^3)^2 = 8^2 = 64

Correction: Exponentiation is right-associative. Always evaluate from the top down:

2^{3^2} = 2^{(3^2)} = 2^9 = 512

. The grouping

(2^3)^2

is a different expression entirely.