2002 AMC 12A Problems/Problem 3

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Problem

According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]

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

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 3\qquad \mathrm{(E) \ } 4$


Solution

Note that $2^{2^2}$ has a unique value of $16$, because $2^4 = 4^2 = 16$

So $2^{2^{2^2}}$ can be perenthesized as either $2^({2^2^2))=2^16$ (Error compiling LaTeX. ! Double superscript.) or $(2^2^2)^2=16^2$ (Error compiling LaTeX. ! Double superscript.)

Therefore, there is one other possible value of $2^2^2^2 \Rightarrow \mathrm {(B)}$ (Error compiling LaTeX. ! Double superscript.)

See Also

2002 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
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All AMC 12 Problems and Solutions
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