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# R = 3^81, R^R = 3^S, S = ?

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Manager
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R = 3^81, R^R = 3^S, S = ? [#permalink]

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13 Apr 2010, 14:23
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Difficulty:

55% (hard)

Question Stats:

62% (01:16) correct 38% (01:28) wrong based on 418 sessions

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$$R = 3^{81}$$

$$R^R = 3^S$$

$$S = ?$$

A. 3
B. 81
C. 3^81
D. 3^85
E. 3^87
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 Aug 2012, 01:52, edited 1 time in total.
Edited the question and added the OA.

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Re: 700+ Level exponent question [#permalink]

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13 Apr 2010, 15:23
not really sure I'm doing this right but anywho...

using simpler numbers if R = 2^2
Then R^R = 2^8

Difference of 6 on the exponents

So 81 + 6 = 87

E

curious if that makes any sense.

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Re: 700+ Level exponent question [#permalink]

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13 Apr 2010, 15:44
well then crap.

no idea.

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Re: 700+ Level exponent question [#permalink]

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13 Apr 2010, 15:48
haha that was a good try though, i thought i had it figured out through that logic... But I still cant figure it out!

Anyone else?

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Re: 700+ Level exponent question [#permalink]

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13 Apr 2010, 18:09
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Don't get freaked by the number of exponents.

((x)^(y))^(z) = (x)^(yz) property

so (3^81)^(3^81) = 3^(81*3^81)

and 3^4 =81

x^y*x^z = x^(y+z)

so 3^(3^4*3^81) = 3^(3^85)

thus s = 3^85

If someone could tell me how to do superscripts I could possibly make this easier to read.

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Re: 700+ Level exponent question [#permalink]

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13 Apr 2010, 19:32
Got it. thanks you. First kudos for you

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Re: 700+ Level exponent question [#permalink]

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13 Apr 2010, 23:37
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$$R = 3^{81}$$

$$R^R = 3^{81*R}$$

$$R^R = 3^{81*3^{81}}$$

$$R^R = 3^{3^4*3{^81}}$$

$$R^R = 3^{3^{4+81}}$$

$$R^R = 3^{3^{85}}$$

thus S = 85
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Re: 700+ Level exponent question [#permalink]

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15 Aug 2012, 08:46
This problem can be solved very quickly using logarithm (although it is not in the syllabus of GMAT)
R=3^81
log (R) = 81 log 3
And since,
R^R = 3^s
therefore, R log R = s log 3
Now substituting the value of log R from eqn. 1,
R X (81 log 3) = s log 3
Hence, s = 3^81 X 3^4
= 3^ 85

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Re: 700+ Level exponent question [#permalink]

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15 Aug 2012, 23:18
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One more way - Exponential -
Given - R = 3^81.....(a)
R^R = 3^S
R = 3 ^(S/R)............(b)
(a) = (b)
we get S/R = 81 i.e. S=81R i.e. S=81 * 3^81
hence S = 3^4 + 3^81 = 3^85
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Re: R = 3^81, R^R = 3^S, S = ? [#permalink]

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27 Feb 2014, 20:55
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Given that R = 3^81

Powering both sides by R

R^R = 3^81R............... (1)

Already given that R^R = 3^S......... (2)

Equating (1) & (2)

S = 81R
= 3^4 . 3^81
= 3^85 = Answer = D
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Re: R = 3^81, R^R = 3^S, S = ? [#permalink]

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28 Feb 2014, 22:36
here's another way....

R^R = (3^81)^(3^81) -->
(27^27)^(27^27)
(27)^(27^28)
(3^3)^[3^(28*3)]
3^[3^(1+28*3)]
3^(3^85)
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Re: R = 3^81, R^R = 3^S, S = ? [#permalink]

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03 Oct 2015, 03:44
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R = 3^81, R^R = 3^S, S = ? [#permalink]

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03 Oct 2015, 07:43
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R=3^81 -----given

R^R=3^S ---given

=> (3^81)^R = 3^S
=> 3^(81R) = 3^S
=> 81R = 3^S
=> 3^4 x 3^81 = 3^S
=> s=52

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Re: R = 3^81, R^R = 3^S, S = ? [#permalink]

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07 Nov 2016, 07:58
Hello from the GMAT Club BumpBot!

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Re: R = 3^81, R^R = 3^S, S = ? [#permalink]

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30 Nov 2016, 17:36
Expert's post
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ksharma12 wrote:
$$R = 3^{81}$$

$$R^R = 3^S$$

$$S = ?$$

A. 3
B. 81
C. 3^81
D. 3^85
E. 3^87

$$R = 3^{81}$$

So, $$R^R = (3^{81})^{3^{81}}$$ [replaced R with 3^81]
$$= (3^{3^4})^{3^{81}}$$ [rewrote 81 as 3^4]
$$= 3^{3^{85}}$$ [applied power of power rule AND product rule]

So, $$3^{3^{85}} = 3^S$$
So, $$S = 3^{85}$$

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Re: R = 3^81, R^R = 3^S, S = ?   [#permalink] 30 Nov 2016, 17:36
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