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If $$2^{2^{2^{2}}}=2^{[2^{(2^{2})}]}=2^{(2^{4})}=2^{16}\;,$$Is it possible to reason any notation properties to make calculations any faster? Or everytime I face a multiplication between iterated powers I'll have to:

1st) Solve them from right to left;

2nd) Multiply like normal exponents;

3rd) Revert the result to the iterated power notation (if there are more multiplications of that sort to be made);

?

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The exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as right-associative. When $a$ and $10$ are coprime, we can compute the last $m$ decimal digits of $^{n}a$ using Euler's theorem.

See this link for computing infinite power towers.

The infinite power tower $x↑↑∞ (\text{or } ^∞x)$ converges so long as

$e^{-e} < x < e^{1/e}.$

It can be written in closed form using the Lambert W function.

Tetrations can be extended to complex numbers, see http://www.tetration.org

If you want to compute the tetration in the exponent, wolframalpha has a cool command. For example:

$$ \text{ Power @@ Table[2, {4}]} $$
Yields: $$ 16 $$ As far as I know there are no computing properties of tetrations to make calculations any faster. If you want to make calculations faster, I suggest using a computer =(

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