Difference between revisions of "2010 AMC 8 Problems/Problem 24"
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Then we can rewrite <math>5^{12}</math> as <math>(5^{\frac{3}{2}})^8=(\sqrt{125})^8)</math>. | Then we can rewrite <math>5^{12}</math> as <math>(5^{\frac{3}{2}})^8=(\sqrt{125})^8)</math>. | ||
We take the eighth root of all of these to get <math>{10, 8, \sqrt{125}}</math>. | We take the eighth root of all of these to get <math>{10, 8, \sqrt{125}}</math>. | ||
| − | Obviously, <math>8<10<\sqrt{125}</math>, so the answer is <math>\textbf{(A)}\ 2^{24}<10^8<5^{12}</math>. | + | Obviously, <math>8<10<\sqrt{125}</math>, so the answer is <math>\boxed{\textbf{(A)}\ 2^{24}<10^8<5^{12}}</math>. |
Solution by MathHayden | Solution by MathHayden | ||
Revision as of 09:00, 9 November 2022
Problem
What is the correct ordering of the three numbers, 
, 
, and 
?
Video Solution
https://youtu.be/rQUwNC0gqdg?t=381
Solution 1
Use brute force.
,
, and
.
Therefore, 
is the answer. (Not recommended for the contest and will take forever)
Solution 2
Since all of the exponents are multiples of four, we can simplify the problem by taking the fourth root of each number. Evaluating we get 
, 
, and 
. Since 
, it follows that 
is the correct answer.
Solution 3
First, let us make all exponents equal to 8. Then, it will be easy to order the numbers without doing any computations.
is as fine as it is.
We can rewrite as 
.
Then we can rewrite as 
.
We take the eighth root of all of these to get 
.
Obviously, 
, so the answer is .
Solution by MathHayden
See Also
| 2010 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
