Question

Порiвняйте два числа: 2^2016(степени) та 3^303x4^404x5^505

Answer 1

$3^{303}*4^{404}*5^{505}=3^{3*101}*4^{4*101}*5^{5*101}=$
$=(3^3)^{101}*(4^4)^{101}*(5^5)^{101} =(3^3*4^4*5^5)^{101}=$
$=(3^3*2^{2*4}*5^5)^{101}=(3^3*2^8*5^5)^{101}\ \textgreater \ (2^3*2^8*4^5)^{101}=$
$=(2^3*2^8*2^{10})^{101}=(2^{21})^{101}=(2^{19+2})^{101}=(2^{19})^{101}*2^{2*101}=$
$=(2^{19})^{101}*2^{202}$

$2^{2016}=2^{19*101+97}=(2^{19})^{101}*2^{97}$

$3^{303}*4^{404}*5^{505}\ \textgreater \ (2^{19})^{101}*2^{202}\ \textgreater \ (2^{19})^{101}*2^{97}=2^{2016}$

ответ: $3^{303}*4^{404}*5^{505}\ \textgreater \ 2^{2016}$