Question

Comparati numerele:
m=3 la puterea 454 -3 la puterea 453-3la puterea452
n=5la puterea 340

Answer 1

$m=3^{454}-3^{453}-3^{452}=3^{452}*(3^{2}-3^{1}-3^{0})=3^{452}*(9-3-1)=3^{452}*5$
$n=5^{340}$
Avem de comparat: $3^{452}*5;5^{340}$
Adica avem de comparat: $3^{452};5^{339} \\ 3^{452}=(3^{4})^{113}=81^{113} \\ 5^{339}=(5^{3})^{113}=125^{113}$
Avem succesiv: $81\ \textless \ 125\to81^{113}\ \textless \ 125^{113}\to3^{452}\ \textless \ 5^{339}\to3^{452}*5\ \textless \ 5^{340}\to m\ \textless \ n$