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

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New post 21 Jul 2015, 11:00
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A
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Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

(a) 2^1/2
(b) 3^1/3
(c) 4^1/4
(d) 6^1/6
(e) 12^1/12

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Source: GMAT prep notes (Shiksha GMAT, Ahmadabad)
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Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ? [#permalink]

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New post 21 Jul 2015, 22:14
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GMATnavigator wrote:
Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

(a) 2^1/2
(b) 3^1/3
(c) 4^1/4
(d) 6^1/6
(e) 12^1/12

Kudos if you like :)

Source: GMAT prep notes (Shiksha GMAT, Ahmadabad)


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
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Re: Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ? [#permalink]

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New post 21 Jul 2015, 21:52
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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
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Re: Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ? [#permalink]

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New post 21 Jul 2015, 23:53
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^[1/3]) is the largest.

Ans (B).
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Which among 2^{1/2}, 3^{1/3}....... [#permalink]

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New post 26 Apr 2017, 23:51
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. \(12^{\frac{1}{12}}\)


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Re: Which among 2^{1/2}, 3^{1/3}....... [#permalink]

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New post 26 Apr 2017, 23:52
Reserving this space to post the official solution. :)
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Re: Which among 2^{1/2}, 3^{1/3}....... [#permalink]

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New post 27 Apr 2017, 00:33
1
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
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Re: Which among 2^{1/2}, 3^{1/3}....... [#permalink]

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New post 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.

Answer is B
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Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ? [#permalink]

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New post 17 Aug 2017, 09:08
GMATnavigator wrote:
Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

(a) 2^1/2
(b) 3^1/3
(c) 4^1/4
(d) 6^1/6
(e) 12^1/12

Kudos if you like :)

Source: GMAT prep notes (Shiksha GMAT, Ahmadabad)


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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Re: Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ? [#permalink]

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New post 17 Aug 2017, 09:40
devarshi9283 wrote:
GMATnavigator wrote:
Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

(a) 2^1/2
(b) 3^1/3
(c) 4^1/4
(d) 6^1/6
(e) 12^1/12

Kudos if you like :)

Source: GMAT prep notes (Shiksha GMAT, Ahmadabad)


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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Re: Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ? [#permalink]

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New post 17 Aug 2017, 10:32
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
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Re: Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?   [#permalink] 17 Aug 2017, 10:32
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