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Rank the following quantities in order, from smallest to biggest. I 2

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Rank the following quantities in order, from smallest to biggest. I 2 [#permalink]

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Rank the following quantities in order, from smallest to biggest.

I. \(2^{120}\)

II. \(3^{72}\)

III. \(17^{30}\)

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

For the OE for this problem, as well as a 11 other challenging problems in exponents & powers & roots, see:
http://magoosh.com/gmat/2014/challengin ... and-roots/

Mike :-)
[Reveal] Spoiler: OA

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Rank the following quantities in order, from smallest to biggest. I 2 [#permalink]

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I. \(2^{120}\)

II. \(3^{72}\)

III. \(17^{30}\)


Let's simplify!
We can't uniform the bases, but it's obvious that exponents can be simplified (they are all multiples of 6).

First,
\(2^{120} = (2^{20})^6\)
\(3^{72}=(3^{12})^6\)
\(17^{30}=(17^5)^6\)

ok, this didn't help much, but let's try to compare only 2 at a time:

\(2^{120} = (2^{20})^6 = (2^{5})^{24} = 32^{24}\)
\(3^{72}=(3^{12})^6 = (3^{3})^{24} = 27^{24}\)

Clearly, I > II

Great! Now, lets compare first and third (120 and 30 are easier).

\(2^{120} = (2^{4})^{30} = 16^{30}\), clearly < \(17^{30}\)

Hence, III > I > II. Answer:
[Reveal] Spoiler:
C
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Re: Rank the following quantities in order, from smallest to biggest. I 2 [#permalink]

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New post 17 Sep 2016, 20:22
We need to compare \(2^{12}\), \(3^{72}\) and \(17^{30}\)

The exponents 120, 72 and 30 have 6 as common factor.

rewriting, \((2^{20}){6}\), \((2^{12}){6}\), \((17^{5}){6}\)

the last term: \((2^4 +1 )^{5}){6}\)
\((2^{20} + 5 *2^{14} +......){6}\)
This is definitely greater than the first term..Second term is the smallest.

Hence, III > I > II
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Re: Rank the following quantities in order, from smallest to biggest. I 2 [#permalink]

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New post 10 Dec 2017, 13:15
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Re: Rank the following quantities in order, from smallest to biggest. I 2   [#permalink] 10 Dec 2017, 13:15
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