Question

Aflati cate cifre are numarul 2³⁰ scris ca numar natural.
Indicatie : Se arata ca 10⁹ < 2³⁰ < 10¹⁰.

Answer 1

$1.~Demonstram~ca~10^9\ \textless \ 2^{30}. \\ \\ 10^9\ \textless \ 2^{30} \Leftrightarrow 2^9 \cdot 5^9\ \textless \ 2^9 \cdot2^{21} \Leftrightarrow 5^9\ \textless \ 2^{21} \Leftrightarrow 125^3\ \textless \ 129^3,adevarat \Rightarrow~ \\ \\ \Rightarrow 10^9\ \textless \ 2^{30}. \\ \\ 2.~Demonstram~ca~2^{30}\ \textless \ 10^{10}.\\ \\2^{30}\ \textless \ 10^{10} \Leftrightarrow 2^{10} \cdot 2^{20}\ \textless \ 2^{10} \cdot 5^{10} \Leftrightarrow 2^{20}\ \textless \ 5^{10} \Leftrightarrow 4^{10}\ \textless \ 5^{10},adevarat \Rightarrow \\ \\ \Rightarrow 2^{30}\ \textless \ 10^{10}.$

$Avem~deci~ca~10^9\ \textless \ 2^{30}\ \textless \ 10^{10},~iar~cum~10^9~are~10~cifre~si~10^{10} \\ \\ are~11~cifre,~rezulta~ca~\boxed{numarul~2^{30}~are~10~cifre.}$