# Parentheses in Power Tower

Algebra Level 2

Suppose we start with a number $2$. Trivially, no matter the number of parentheses, the result is still $2$.

Squaring that, we have $2^2$. Then, the results are the same for any choice of parentheses.

For three $2$'s, wherever the parentheses are set, the result is $16$. For instance, $(2^2)^2 = 2^{(2^2)}$.

What about the power tower of at least four $2$'s? Would the result still be the same for any choice of parentheses and for all number of $2$'s?

Note: Order of operations holds.

×