수학의 재미 2010 AMC 8 문제 24 어느수가 더 클까요?

Problem

 

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

문제​ )

10^8​ , 5^12 , 2^24 세수의 올바른 순서는 다음의 어느것 인가?

Solution 1

Use brute force. , 

,

and . Therefore,

 

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 , , 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 fine as is. We can rewrite  as . We can rewrite  as . We take the eighth root of all of these to get . Obviously, 

 so the answer is 

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

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