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Rank those three in order from smallest to biggest.

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Magoosh GMAT Instructor
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Rank those three in order from smallest to biggest. [#permalink]

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New post 27 Sep 2016, 17:20
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61% (01:06) correct 39% (01:01) wrong based on 129 sessions

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Attachment:
three roots.png
three roots.png [ 34.21 KiB | Viewed 1174 times ]

Rank those three in order from smallest to biggest.

(A) I, II, III
(B) I, III, II
(C) II, I, III
(D) II, III, I
(E) III, II, I


This problem involves a bit of number sense. For a discussion of this skill, as well as the OE for this particular question, see:
Number Sense for the GMAT

Mike :-)
[Reveal] Spoiler: OA

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Re: Rank those three in order from smallest to biggest. [#permalink]

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New post 27 Sep 2016, 18:45
We can write all the terms with the same exponent
\(\sqrt{5}\) , \(\sqrt[3]{2}\), \(\sqrt[4]{401}\)
\(\sqrt[3]{5^6}\), \(\sqrt[3]{2^4}\), \(\sqrt[4]{401^3}\)

We know \(2^4\) < \(5^6\), so II < I

\(5^6\) = \(125^3\) <\(401^3\)
So I < III

II < I < III
Option C
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Re: Rank those three in order from smallest to biggest. [#permalink]

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New post 27 Sep 2016, 19:44
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mikemcgarry wrote:
Attachment:
three roots.png

Rank those three in order from smallest to biggest.
(A) I, II, III
(B) I, III, II
(C) II, I, III
(D) II, III, I
(E) III, II, I


This problem involves a bit of number sense. For a discussion of this skill, as well as the OE for this particular question, see:
Number Sense for the GMAT

Mike :-)


Get them in same roots..
I. 2√5= √(2*2*5)=√20= 4th root of 20*20 or 4th root of 400
II. 3√2=√3*3*2=√18 ---- so LESS than I

Let's check III now
III. 4th root of 401-------- so GREATER than I.. therefore the greatest

So smallest to largest ----- II, I, III

C
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Re: Rank those three in order from smallest to biggest. [#permalink]

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New post 27 Sep 2016, 20:55
for expressions like this we need to take lcm of the roots of the expressions

on taking lcm, these terms can be written as 5^6 , 2^4, 401^3

now its clear which one is smallest and which one is biggest term

2^4 < 5^6 < 401^3

Option C
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Re: Rank those three in order from smallest to biggest. [#permalink]

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New post 24 Dec 2017, 07:02
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Raising the power of all the options to power 4, will help rank easily

1. \((2 * \sqrt{5})^4\) = \(2^4 * 5^2\) = 400
2. \((3 * \sqrt{2}) ^ 4\) = \(3^4 * 2 ^ 2\) = 81 * 4 = 324
3. \((401 ^ {1/4})^4\) = 401

Rank from smallest to biggest : 2, 1, 3

Answer (C)
Re: Rank those three in order from smallest to biggest.   [#permalink] 24 Dec 2017, 07:02
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