2010 AMC 8 Problems/Problem 24

Problem

What is the correct ordering of the three numbers, , , and ?

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

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 , , and . Since , it follows that $\boxed{\textbf{(A)}\ 2^2^4<10^8<5^1^2 }$ (Error compiling LaTeX. Unknown error_msg) is the correct answer.