Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

Question

Which among  2^{(\frac{1}{2})}, \ 3^{(\frac{1}{3})}, \ 4^{(\frac{1}{4})}, \ 6^{(\frac{1}{6})}  and  12^{(\frac{1}{12})}  is the largest ?

(A)  2^{(\frac{1}{2})}

(B)  3^{(\frac{1}{3})}

(C)  4^{(\frac{1}{4})}

(D)  6^{(\frac{1}{6})}

(E)  12^{(\frac{1}{12})}

Source: GMAT prep notes (Shiksha GMAT, Ahmadabad)

GMATinsight · Accepted Answer

CONCEPT: The comparison of Difficult Surds and Indices can be done when either their bases are equal or their powers are equal.

Here the bases of numbers can't be made same so making their powers equal

2^{1/2} ,  3^{1/3} ,  4^{1/4}  ,  6^{1/6}  and  12^{1/12}

2^{1/2}  =  (2^6)^{1/12}  =  64^{1/12}  
  3^{1/3}  =  (3^4)^{1/12}  =  81^{1/12}   LARGEST   
 4^{1/4}  =  (4^3)^{1/12}  =  64^{1/12}  
 6^{1/6}  =  (6^2)^{1/12}  =  36^{1/12}  
 12^{1/12}  =  12^{1/12}  =  12^{1/12}

Answer: option B

Mesto · Answer

2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12

Multiply each number raised to the power of 12
So,

2^1/2 = 2^6 = 64
3^1/3 = 3^4 = 81
4^1/4 = 4^3 = 64
6^1/6 = 6^2 = 36
12^1/12 = 12^1 = 12

So, the greatest is 3^1/3

TimeTraveller · Answer

2^(1/2), 3^(1/3), 4^(1/4), 6^(1/6), 12^(1/12).

LCM(2,3,4,6,12) = 24.

Now, let's compare 2^(24/2), 3^(24/3), 4^(24/4), 6^(24/6), 12^(24/12).

=> 2^(12), 3^(8), 4^(6), 6^(4), 12^(2).

=> 2^(12), 3^(8), 2^(12), (3^4 * 2^4), (144).

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^ ) is the largest.

Ans (B).

EgmatQuantExpert · Answer

Which among 2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, 6^{\frac{1}{6}}, 12^{\frac{1}{12}} is the largest? A. 2^{\frac{1}{2}} B. 3^{\frac{1}{3}} C. 4^{\frac{1}{4}} D. 6^{\frac{1}{6}} E. 12^{\frac{1}{12}} Thanks, Saquib Quant Expert e-GMAT Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts

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VyshakhR1995 · Answer

2^(1/2)
 3^(1/3 ) 
4^(1/4)
 6^(1/6)
 12^(1/12)
Are the given numbers 
Raising each term to the power of 12

(2)^6 (3)^4 (4)^3 (6)^2 (12)^1

Clearly (3)^4 is largest among the numbers 
------->3^(1/3 ) largest
B

quantumliner · Answer

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.

Answer is B

devarshi9283 · Answer

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

GMATinsight · Answer

Logarithms is not part of GMAT hence GMAT specific readers may avoid any solution concerning Logarithms (log function)

Posted from my mobile device

sahilvijay · Answer

Answer is B clearly

make multiply all by 12th power

new numbers will become 2^6, 3^4, 4^3, 6^2 , 12
64,81,64,36,12

clearly 81 is biggest number.

hence B

findingmyself · Answer

A simple concept that you can keep in mind: 32^1/5=2, 64^1/6=2 higher the root to be determined (1/6,1/12,1/48), no matter how big the number is, root will be small. or in other words 12^1/12 is a number which when multiplied 12 times gives 12.

Such a number will always be lower than \sqrt{}2 and \sqrt{}3.

Now we can see that 4^1/4 is nothing but \sqrt{}2 only.

Between \sqrt{}2 and \sqrt{}3, \sqrt{}3 will always be higher

Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

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GMAT Problem Solving (PS) Questions

Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

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