Question

chứng minh 2^116<10^35

Answer 1

ta có:25.125^11 < 128^11.25 < 2^77.32 = 2^82 
=> 25.125^11 < 2^82 
=> 5^35 < 2^82 
=> 10^35 < 2^117 

ta có: 
2^96 = 4096^8 
2^96 < 41^8.10^16 
2^81 < 2.41^8.5^16...(*) 

lại có: 9.2^13 < 9.8200 < 73000 < 625.125 
=> 9.2^13 < 5^7 
=> 300^2.2^9 < 5^11 
=> 17^4.2^9 < 5^11...(vì 17^2 <300) 
=> 1700^4.2 < 5^19 
=> 2.41^8 < 5^19 ...(vì 41^2 <1700) 
=> 2.41^8.5^16 < 5^35 
kết hợp với (*) => 2^81 < 5^35 

=> 2^81 < 5^35 < 2^81 
=> 2^116 < 10^35 < 2^117....đpcm 

Answer 2

\(10^{35}=2^{35}.5^{35}< 2^{35}.4^{35}=2^{35}.\left(2^2\right)^{35}=2^{35}.2^{70}=2^{105}.\)

\(2^{116}>2^{105}\)

\(\Rightarrow2^{116}>10^{35}\)