2002 AMC 10A Problem 3

Question

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?

0

1

2

3

4

List the five ways to parenthesize a tower of four

2

’s

Every grouping evaluates to either

2^{16}

or

2^{8}

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.

Problem 3 in other years