2002 AMC 10A Problem 3

According to the standard convention for exponentiation,

2^{2^{2^{2}}}=2^{\left(2^{\left(2^{2}\right)}\right)}=2^{16}=65{,}536.

If the order in which the exponentiations are performed is changed, how many other values are possible?

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Solution

There are five ways to parenthesize the tower. Three of them,

(2^2)^{\left(2^2\right)},

\left(2^{\left(2^2\right)}\right)^2,

and

\left(\left(2^2\right)^2\right)^2,

all equal

2^{8}=256.

The other two both give the standard value

2^{16}=65{,}536.

So exactly one other value,

256,

is possible. Thus, the correct answer is B.