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.

Yes No

1 solution

Brian Charlesworth
Mar 28, 2017

$\Large (2^{2})^{(2^{2})} = 4^{4} = 2^{8} = 256$ , but

$\Large 2^{(2^{(2^{2})})} = 2^{(2^{4})} = 2^{16} = 65536$ .

So the answer is $\boxed{\text{No}}$ .