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

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

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New post Updated on: 16 Aug 2012, 01:52
2
15
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

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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

Originally posted by ksharma12 on 13 Apr 2010, 14:23.
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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New post 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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New post 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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New post 13 Apr 2010, 18:09
3
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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New post 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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New post 15 Aug 2012, 23:18
2
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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New post 27 Feb 2014, 20:55
1
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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New post 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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R = 3^81, R^R = 3^S, S = ?  [#permalink]

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New post 03 Oct 2015, 07:43
1
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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New post 30 Nov 2016, 17:36
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Top Contributor
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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, \(S = 3^{85}\)

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

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New post 09 Apr 2019, 04:41
R=3^81

R^R=3^S

(3^81)^R = 3^S

So, 3^(81R) = 3^S

81R = S

=> S=(3^4)(3^81)

S=3^85

(D)
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Re: R = 3^81, R^R = 3^S, S = ?   [#permalink] 09 Apr 2019, 04:41
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