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

Question

R = 3^{81}

R^R = 3^S

S = ?

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

gurpreetsingh · Accepted Answer

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

BrentGMATPrepNow · Answer

R = 3^{81}

So,  R^R = (3^{81})^{3^{81}}     
 = (3^{3^4})^{3^{81}}     
 = 3^{3^{85}}

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

Answer: D

run4run · Answer

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.

GVJ81 · Answer

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.

sourabh123 · Answer

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

methevoid · Answer

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

PareshGmat · Answer

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

m3equals333 · Answer

here's another way....

R^R = (3^81)^(3^81) -->
(27^27)^(27^27)
(27)^(27^28)
(3^3)^ 
3^ 
3^(3^85)

draditya · Answer

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

firas92 · Answer

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)

kubrah · Answer

What is this concept called exactly and are there any videos on youtube explaining the principles?

Kind regards.

Bunuel · Answer

8. Exponents and Roots of Numbers 
     Theory   
 Exponents and Roots on the GMAT: Tips and hints 
 Cyclicity and the remainders on the GMAT 
     Questions   
 Units digits, exponents, remainders problems 
 Tough and tricky DS exponents and roots questions with detailed solutions 
 Tough and tricky PS exponents and roots questions with detailed solutions 
 Exponents PS Questions 
 Roots DS Questions 
 Roots PS Questions

Check below for more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope this helps.

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

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards