Question

Arătați că 2^1000,scris in baza 10 ,are cel puțin 301 cifre? ​

Answer 1

 

$\displaystyle\bf\\2^{1000}=2^{10\times100}=\Big(2^{10}\Big)^{100}=\boxed{\bf1024^{100}}\\\\1000<1024\\\\1000^{100}<1024^{100}\\\\1000^{100}=\Big(10^3\Big)^{100}=10^{3\times100}=10^{300}\\\\10^{300}=1\underbrace{000000....000000}_{300~de~zerouri}\\\\10^{300}~are~ un~"1" + 300~de~zerouri=\boxed{\bf301~cifre}\\\\\implies~1000^{100}~are~301~cifre\\\\1024^{100}>1000^{100}\\\\\implies~1024^{100}~are~cel~putin~301~cifre\\\\\implies~\boxed{\bf2^{1000}~are~cel~putin~301~cifre}$