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We only need to compare 3^(8) and (3^4 * 2^4). (3^8) = (3^4 * 3*4), which is clearly greater than (3^4 * 2^4). Therefore, 3^8 (=3^[1/3]) is the largest.
Ans (B).
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Re: Which among 2^{1/2}, 3^{1/3}....... [#permalink]
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27 Apr 2017, 00:46
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As the bases are different, we can try to make their powers equal.
2^(1/2) can be written as (2^6) ^ (1/12) = 64 ^ (1/12) 3^(1/3) can be written as (3^4) ^ (1/12) = 81 ^ (1/12) 4^(1/4) can be written as (4^3) ^ (1/12) = 64 ^ (1/12) 6^(1/6) can be written as (6^2) ^ (1/12) = 36 ^ (1/12) 12^(1/12) can be written as (12^1) ^ (1/12) = 12 ^ (1/12)
Clearly we can see that 3^(1/3) is the largest of all.
Although method elaborated earlier is the easiest and the quickest, I will put forward "Log method" for such questions too...
take Log of all numbers given.. log 2^1/2 becomes 1/2 log 2 log 3^1/3 becomes 1/3 log 3 log 4^1/4 becomes 1/4 log 4 which becomes 1/4 log 2^2 which becomes 1/2 log 2 (A and C equal hence eliminated) log 6^1/6 becomes 1/6 log 6 which becomes 1/6 log (2*3) which becomes 1/6 (log 2+log 3) which is less than 1/3 (log3) as 1/3 log3 = 1/6 (log 3+log3) (D eliminated) log 12^1/12 becomes 1/12 log 12 which becomes 1/12 log (2^2*3) which becomes 1/12 {2log2+log3} which is less than 1/3 log 3 as 1/3 log 3 =1/12 {2log3+2log3} (E eliminated) Answer= B
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Although method elaborated earlier is the easiest and the quickest, I will put forward "Log method" for such questions too...
take Log of all numbers given.. log 2^1/2 becomes 1/2 log 2 log 3^1/3 becomes 1/3 log 3 log 4^1/4 becomes 1/4 log 4 which becomes 1/4 log 2^2 which becomes 1/2 log 2 (A and C equal hence eliminated) log 6^1/6 becomes 1/6 log 6 which becomes 1/6 log (2*3) which becomes 1/6 (log 2+log 3) which is less than 1/3 (log3) as 1/3 log3 = 1/6 (log 3+log3) (D eliminated) log 12^1/12 becomes 1/12 log 12 which becomes 1/12 log (2^2*3) which becomes 1/12 {2log2+log3} which is less than 1/3 log 3 as 1/3 log 3 =1/12 {2log3+2log3} (E eliminated) Answer= B
Logarithms is not part of GMAT hence GMAT specific readers may avoid any solution concerning Logarithms (log function)
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48% (01:03) correct 
