Question

Problema 3. (7 puncte)
a) Demonstrați că 21²² < 5³³×2²²
b) Aflați care dintre numerele a = 2²³×5³⁵ şi b = 3²²×7²⁴ este mai mare.​

Answer 1

a)

·$21^{22} < 5^{33} 2^{22} \\21^{22} < 5^{22} 5^{11} 2^{22}$

$(\frac{21}{5} )^{22} < 5^{11} 2^{22}$

$(\frac{21}{10} )^{22} < 5^{11}$

$(\frac{21}{10} )^{11} (\frac{21}{10} )^{11} <5^{11}$

Imparti relatia la 5^11 , rezultand :

$(\frac{21}{10} )^{11} (\frac{21}{10} )^{11} \frac{1}{5} ^{11} < 1$

$(\frac{21}{10} )^{11} (\frac{21}{50} )^{11} < 1$

$(\frac{441 }{500} )^{11} < 1$

$\frac{441}{500} < 1\\ 441 < 500$

b) Demonstram prin absurd ca a < b .

$2^{23} 5^{35} <3^{22} 7^{24}$

$2^{22} *2*5^{22} *5^{13} < 3^{22} * 7^{22} * 7^{2}$

$(\frac{2*5}{3*7} )^{22} * 2 < 7^{2}$

$(\frac{2*5}{3*7} )^{22} < \frac{ 49 }{2}$

$(\frac{10}{21} )^{22} < \frac{ 49 }{2}$

Cum 10 / 21 < 1 inseamna ca orice numar subunitar ridicat la o putere tinde sa se micsoreze(tinde spre zero).

Astfel relatia $(\frac{10}{21} )^{22} < \frac{ 49 }{2}$ e adevarata => a < b