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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

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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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]

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