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# Power Stack

$$

3^{(3^3)} = 3^{(27)} = 7625597484987\quad\quad (3^3)^3 = 27^3 = 19683

$$

The difference rapidly grows for larger values:

$$ 4^{(4^4)} = 4^{(256)} \sim 10^{154} \quad\quad (4^4)^4 = 256^4\sim 10^9 $$

However, for $2$ the values are the same

$$ 2^{(2^2)} = 2^{(4)} = 16\quad\quad (2^2)^2 =4^2 =16 $$

The extension of the definitions are naturally either 'powers evaluated from the right' or 'powers evaluated from the left'. The difference for a stack of four powers is gigantic

$$

(((3^3)^3)^3) = (((27)^3)^3) = (19683)^3\sim 10^{12}

$$

$$

(3^{(3^{(3^{(3)})})}) =(3^{(3^{27})}) =(3^{(7.6\times 10^{12})})\sim 10^{3.6\times 10^{12}}

$$

Using a spreadsheet we found that both definition of stacking four numbers leads to the same value when the base is $1.02092370325178$

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$$

3^{(3^3)} = 3^{(27)} = 7625597484987\quad\quad (3^3)^3 = 27^3 = 19683

$$

The difference rapidly grows for larger values:

$$ 4^{(4^4)} = 4^{(256)} \sim 10^{154} \quad\quad (4^4)^4 = 256^4\sim 10^9 $$

However, for $2$ the values are the same

$$ 2^{(2^2)} = 2^{(4)} = 16\quad\quad (2^2)^2 =4^2 =16 $$

The extension of the definitions are naturally either 'powers evaluated from the right' or 'powers evaluated from the left'. The difference for a stack of four powers is gigantic

$$

(((3^3)^3)^3) = (((27)^3)^3) = (19683)^3\sim 10^{12}

$$

$$

(3^{(3^{(3^{(3)})})}) =(3^{(3^{27})}) =(3^{(7.6\times 10^{12})})\sim 10^{3.6\times 10^{12}}

$$

Using a spreadsheet we found that both definition of stacking four numbers leads to the same value when the base is $1.02092370325178$