Question

Care este mai mare 3^1275 sau 5^465?

Answer 1

$Observam~ca~ 3^{1275}=\left(3^{85} \right)^{15}~si~5^{465}= \left( 5 ^{31}\right)^{15}. \\ \\ Deci~a~compara~numerele~\left(3^{85} \right)^{15}~si~ \left( 5 ^{31}\right)^{15} ~este~echivalent~cu~a \\ \\ compara~numerele~3^{85}~si~5^{31}. \\ \\ Avem:~3^{85}=\left(3^5 \right)^{17}=243^{17}=243^2 \cdot 243^{15} \geq 5 \cdot 25^{15}=5 \cdot 5^{30}= \\ \\ =5^{31}. \\ \\ Deci~3^{85}\ \textgreater \ 5^{31},~ceea~ce~inseamna~ca~3^{1275}\ \textgreater \ 5^{465}.$

$Solutie~alternativa:~3^{1275}=9^{637,5}\ \textgreater \ 5^{637,5}\ \textgreater \ 5^{465}.$