Difference between revisions of "2002 AMC 12A Problems/Problem 3"
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If the order in which the exponentiations are performed is changed, how many other values are possible? | If the order in which the exponentiations are performed is changed, how many other values are possible? | ||
| − | <math> \ | + | <math> \textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 2\qquad \textbf{(D) } 3\qquad \textbf{(E) } 4 </math> |
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==Solution== | ==Solution== | ||
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<math>(2\uparrow 2)\uparrow (2\uparrow 2) = 4 \uparrow 4 = 256</math> | <math>(2\uparrow 2)\uparrow (2\uparrow 2) = 4 \uparrow 4 = 256</math> | ||
| − | Thus the only other result is <math>256</math>, and our answer is <math>\boxed{\ | + | Thus the only other result is <math>256</math>, and our answer is <math>\boxed{\textbf{(B) } 1}</math>. |
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| + | ==Video Solution by Daily Dose of Math== | ||
| + | |||
| + | https://youtu.be/fJndjYHWBrU | ||
| + | |||
| + | ~Thesmartgreekmathdude | ||
==See Also== | ==See Also== | ||
Latest revision as of 23:42, 18 July 2024
- The following problem is from both the 2002 AMC 12A #3 and 2002 AMC 10A #3, so both problems redirect to this page.
Problem
According to the standard convention for exponentiation,
![\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]](http://latex.artofproblemsolving.com/2/9/d/29dee8eac6c7812722f0013ad002044b30d8f220.png)
If the order in which the exponentiations are performed is changed, how many other values are possible?
Solution
The best way to solve this problem is by simple brute force.
It is convenient to drop the usual way how exponentiation is denoted, and to write the formula as 
, where 
denotes exponentiation. We are now examining all ways to add parentheses to this expression. There are 5 ways to do so:
We can note that 
. Therefore options 1 and 2 are equal, and options 3 and 4 are equal.
Option 1 is the one given in the problem statement. Thus we only need to evaluate options 3 and 5.
Thus the only other result is 
, and our answer is 
.
Video Solution by Daily Dose of Math
~Thesmartgreekmathdude
See Also
| 2002 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 2 |
Followed by Problem 4 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
| 2002 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
