"Faster" version of powers.

Question

I know that essentially, multiplication is just a faster version of addition, as $5 \times 3$ is just $5 + 5 + 5$. I also know that powers are a faster version of multiplication as $5^3$ is $5 \times 5 \times 5$. I am wondering if there is a faster version of powers such as $5$ insert symbol $3$ is $5^{(5^5)}$.

Answer 1

The operation you describe is called tetration and is sometimes denoted with a leading superscript. For instance, $$ {}^3 5 = 5^{5^5} \text{.} $$ More generally, for real $a \neq 0$ and nonnegative integer $n$, $$ {}^n a = \begin{cases} 1 ,& n = 0 \\ a^{\left( {}^{(n-1)} a \right)} ,& n > 0 \text{.} \end{cases} $$

The generalization you are starting along is the hyperoperation sequence. Continuing on, each operation is just repeated application of the previous operation. Tetration is repeated exponentiation, pentation is repeated tetration, and so on. (The notation for hyperoperations is ... awful, but they don't usually come up, so there is little need to make it better.)