# Tetration

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Tetration is an operation used in googology.

## Why Tetration?

Googology is the study of large finite numbers. Mathematicians sometimes need new operations to write these numbers down. Some numbers, you can write with exponentiation. For example, a million is written by $10^6,$ and a googol is written with a $10^{100}.$ What about googolplex, $10^{10^{100}}?$ It can still be written with exponentiation. Let's say we invented a new number that had the value of $\[10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}.\]$ Whoa. I don't think you would want to write ten 10's every time!

## Fundamentals

We all know that multiplication is repeated addition. We learned that in elementary school. The definition of $ab$ is $a+a+a+\cdots+a+a$ with $b$ $a$'s.

We also know that exponentiation is repeated multiplication.

Have you ever wondered what was repeated exponentiation?

Yep, you guessed it. Tetration is repeated exponentiation.

## Explanation

In order to write $\[10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}\]$ with tetration, you would write $10 \uparrow \uparrow 10,$ because there are ten 10's. Similarly, $3 \uparrow \uparrow 2$ would be $3^3 = 27,$ (there are 2 three's) and $3 \uparrow \uparrow 3$ would equal $3^{27},$ which is a really big number (around 7 trillion)!

## What's Next?

Now, are you wondering what repeated tetration is? Click on the link to find out!

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