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?

$0$

$1$

$2$

$3$

$4$

Answer: B

Concepts:exponent order of operations systematic listing

Difficulty rating: 1190

Small Hint:

List the five ways to parenthesize a tower of four $2$ ’s

Big Hint:

Every grouping evaluates to either $2^{16}$ or $2^{8}$

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.

2002 AMC 10A Problem 3

Answer: B

Difficulty rating: 1190

Small Hint:

Big Hint:

Solution:

Problem 3 in Other Years