Difference between revisions of "2010 AMC 8 Problems/Problem 24"

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Revision as of 00:58, 5 July 2013

Problem

What is the correct ordering of the three numbers, $10^8$, $5^{12}$, and $2^{24}$?

$\textbf{(A)}\ 2^2^4<10^8<5^1^2$ (Error compiling LaTeX. ! Double superscript.) $\textbf{(B)}\ 2^2^4<5^1^2<10^8$ (Error compiling LaTeX. ! Double superscript.) $\textbf{(C)}\ 5^1^2<2^2^4<10^8$ (Error compiling LaTeX. ! Double superscript.) $\textbf{(D)}\ 10^8<5^1^2<2^2^4$ (Error compiling LaTeX. ! Double superscript.) $\textbf{(E)}\ 10^8<2^2^4<5^1^2$ (Error compiling LaTeX. ! Double superscript.)

Solution

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 $10^2=100$, $5^3=125$, and $2^6=64$. Since $64<100<125$, it follows that $\boxed{\textbf{(A)}\ 2^2^4<10^8<5^1^2 }$ (Error compiling LaTeX. ! Double superscript.) is the correct answer.

See Also

2010 AMC 8 (ProblemsAnswer KeyResources)
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

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