Tetration Facts for Kids

Tetration is a super-powerful math operation. It's like the next step after exponentiation (which is repeated multiplication). Imagine you have a number, and you want to raise it to a power, then raise the result to that same power again, and again! That's what tetration does. It's sometimes called "super-exponentiation."

What are Hyperoperations?
Tetration Example
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What are Hyperoperations?

Math has different levels of operations. We start with simple ones and build up to more complex ones. These are called hyperoperations. Think of them as a family of operations, each one building on the last.

Level 1: Addition

Addition is the most basic. It's about combining numbers.
For example, 3 + 3 + 3 means you add 3 to itself two more times.
You can write this as: $a + n = a\!\underbrace{''{}^{\cdots}{}'}_n$
This means a is increased n times.

Level 2: Multiplication

Multiplication is like a shortcut for repeated addition.
Instead of 3 + 3 + 3, you can write 3 × 3.
You can write this as: $a \times n = \underbrace{a + a + \cdots + a}_n$
This means a is added to itself, n times.

Level 3: Exponentiation

Exponentiation is a shortcut for repeated multiplication.
Instead of 3 × 3 × 3, you can write 3³.
You can write this as: $a^n = \underbrace{a \times a \times \cdots \times a}_n$
This means a is multiplied by itself, n times.

Level 4: Tetration

Tetration is the next step. It's a shortcut for repeated exponentiation.
It means you take a number and raise it to its own power, then take that result and raise it to the power of the original number again, and so on.
The symbol for tetration looks like this: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ^{n}a .
You can write this as: ${^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n$
This means a is exponentiated by itself, n-1 times.
The way you read Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{n}a} is "the nth tetration of a."

Tetration Example

Let's look at an example of tetration. We will calculate Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{2}3} .

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{2}3} means we take the number 3 and raise it to the power of 3.
So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{2}3} = 3^3 .
Remember, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3^3 means $3 \times 3 \times 3$ .
First, $3 \times 3 = 9$ .
Then, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 9 \times 3 = 27 .
So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{2}3} = 27 .

Let's try another one, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{3}2} .

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{3}2} means Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^{2^2} .
We work from the top down. First, calculate the top exponent: $2^2 = 4$ .
Now, the expression becomes $2^4$ .
$2^4$ means Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \times 2 \times 2 \times 2 .
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \times 2 = 4
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 4 \times 2 = 8
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 8 \times 2 = 16
So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {^{3}2} = 16 .