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Roots

by myfish » Tue Jun 19, 2012 7:58 pm
I guessed this one correct but I would love to know an under 2min solution. Thanks so much.
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by eagleeye » Tue Jun 19, 2012 8:16 pm
Hi myfish:

You can do it easily in 10 seconds if you observe the following exponent property:

1) sqrt(4) = 2
2) (number greater than 1)^(positive number) > 1.
In this case, 4 is greater than 1. Now,
The smallest value of (4)^(positive number) > 4^0 ,
And we know 4^0 = 1, hence both 4^(1/3) and 4^(1/4) are greater than 1.

So sum = 4^(1/2)+4^(1/3)+4^(1/4) > 2 + 1 + 1
=> sum > 4.

On a lighter note, this may be the first instance of an "eagle" helping a "fish"

Let me know if this helps :)
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by Anurag@Gurome » Tue Jun 19, 2012 8:51 pm
As 4 > 1,
  • 1 < fourth root of 4 < cube root of 4 < square root of 4
Now, square root of 4 = 2

Therefore, M = 2 + (Some quantity greater than 1) + (Some quantity greater than 1) > (2 + 1 + 1) = 4

Hence, M > 4

The correct answer is E.
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by GMATGuruNY » Wed Jun 20, 2012 2:31 am
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Since the problem above asks only for an estimation, we can ballpark:

√4 = 2.
4^(1/3) > 1.
4^(1/4) = √2 ≈ 1.4.

So M > 4.4.

The correct answer is E.
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by myfish » Thu Jun 21, 2012 8:27 pm
[quote="eagleeye"]Hi myfish:

You can do it easily in 10 seconds if you observe the following exponent property:

1) [b]sqrt(4) = 2[/b]
2)[b] (number greater than 1)^(positive number) > 1[/b].
In this case, 4 is greater than 1. Now,
The smallest value of (4)^(positive number) > 4^0 ,
And we know 4^0 = 1, hence both 4^(1/3) and 4^(1/4) are greater than 1.

So sum = 4^(1/2)+4^(1/3)+4^(1/4) > 2 + 1 + 1
=> sum > 4.

On a lighter note, this may be the first instance of an "eagle" helping a "fish"

Let me know if this helps :)[/quote]

It looks to me the order in nature is undisturbed as the eagle caught the fish wo(a)ndering around!
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