Question

Ajutati-ma va rog frumos la matematica dau Fundita !
1. Aratati ca : 246 la puterea 642>642 la puterea 246
2. Aratati ca : 144la puterea 441 >441 la puterea 144

Answer 1

1)
Se calculeaza cmmdc(642,246)
Pentru aceasta se descompun numerele 246 si 642 in factori primi.
642/2
321/3
107/107
1/
642=2^1*3^1*107^1
246/2
123/3
41/41
1/
246=2^1*3^1*41^1
Se iau toti factorii primi comuni la puterile cele mai mici.
cmmdc(642,246)=2^1*3^1=6
[tex]246^{642}=246^{107*6}=(246^{107})^{6}\\ 642^{246}=642^{41*6}=(642^{41})^{6}\\ A\ compara\ (246^{107})^{6}\ cu\ (642^{41})^{6} este\ echivalent\ cu\ a\ compara\\ 246^{107}\ cu\ 642^{41}\\ 246^{107}=246^{2*53,5}=(246^{2})^{53,5}=60516^{53,5}\ \textgreater \ 642^{41}\\ ccctd[/tex]