2019년 8월 15일 목요일

수학의 재미 2010 AMC 8 문제 24 어느수가 더 클까요?

Problem

 

What is the correct ordering of the three numbers, $10^8$$5^{12}$, and $2^{24}$?
문제​ )
10^8​ , 5^12 , 2^24 세수의 올바른 순서는 다음의 어느것 인가?

$\textbf{(A)}\
$\textbf{(B)}\
$\textbf{(C)}\
$\textbf{(D)}\
$\textbf{(E)}\

Solution 1

Use brute force. $10^8=100,000,000$
$5^{12}=244,140,625$,
and $2^{24}=16,777,216$. Therefore,

 $\boxed{\text{(A)}2^{24}
is the answer.

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 $10^2=100$$5^3=125$, and $2^6=64$. Since $64

it follows that $\boxed{\textbf{(A)}\
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. $10^8$ is fine as is. We can rewrite $2^{24}$ as $(2^3)^8=8^8$. We can rewrite $5^{12}$ as $(5^{\frac{3}{2}})^8=(\sqrt{125})^8)$. We take the eighth root of all of these to get ${10, 8, \sqrt{125}}$. Obviously, $8
 so the answer is $\textbf{(A)}\

풀이 1)
무작정 계산하기
10^8 은 100000000
5^12 는 244140625
2^24 는 16777216
2^24 <  10^8  < 5^12
답은 A

풀이 2)

지수 8, 12 ,24 의 최대공약수가 4 니까
10^8 = ( 10^2)^4 = 100^4
5^12 = ( 5^3 )^4 = 125^4
2^24 = ( 2^6)^4 = 64^4

64 <  100  < 125
니까
2^24 <  10^8  < 5^12
답은 A




문의 는 010-3549-5206 로 하세요

댓글 없음: