Infinite Factorials

$\Large{ x! }^{\color{#D61F06}{{ x! }^{ { x! }^{ \dots } } }}=\color{#D61F06}{1}$ $\Large({ x! })^{\color{#D61F06}{1}}=\color{#D61F06}{1}$ $~~~~~~~~~~~~~~~~\large\color{#3D99F6}{\boxed{ x!=1}}~~(x>0)$

INDIANS ARE BEST MATHEMATICIAN

${ { 1! ^ { 1! } } ^ { 1! } }^ { \cdots } = 1 \Rightarrow x = 0, x = 1$ , but $x > 0$ , so the answer is $\boxed { 1 }$ .

$x!^{x!^{x!^{\dots}}}=1 \implies y=x!^y=1 \implies y=1=x! \implies 1=x! \implies x<0,x=0,1 \implies x=1$

here we know that 1! is the only term which is greater than zero and having its value 1 .

No number except 1 works in this formula since no factorial value equals 0, so the value of the raised factorial powers has to be 1.

No factorial value equals $0$ , so the value of the raised factorial powers has to be $1$ . The only number to the $1$ power which equals $1$ is $1$ , therefore giving us our answer of $1$ .