So sánh \(2^{100}và10^{31}\)
ta có: \(2^{100}=2^{31}.2^{69}\)
\(=2^{31}.2^{63}.2^6\)
\(=2^{31}.\left(2^9\right)^7.\left(2^2\right)^3\)
\(=2^{31}.512^7.4^3\) \(\left(1\right)\)
\(100^{31}=2^{31}.5^{31}\)
\(=2^{31}.5^{28}.5^3\)
\(=2^{31}.\left(5^4\right)^7.5^3\)
\(=2^{31}.625^7.5^3\left(2\right)\)
Từ (1) và (2), ta có:
\(2^{31}.512^7.4^3< 2^{31}.312^7.5^3< 2^{31}.625^7.5^3\)
hay \(2^{100}< 10^{31}\)
Ta có: 2100= 231.269
= 231.266.23
= 231.(29)7.23
= 231.5127.23 (1)
1031= 231.531
= 231.528.53
= 231.(54)7.53
= 231.6257.53 (2)
Từ (1) và (2) suy ra: 231.6257.53>231.5127.53> 231.5127.23
Chúc bạn học tốt!!!
Ta có :
\(2^{100}=2^{31}.2^{69}\)
= \(2^{31}.2^{63}.2^6\)
= \(2^{31}.\left(2^9\right)^7.\left(2^2\right)^3\)
= \(2^{31}.512^7.4^{3^{\left(1\right)}}\)
\(10^{31}\)= \(2^{31}.5^{31}\)
= \(2^{31}.5^{28}.5^3\)
= \(2^{31}.\left(5^4\right)^7.5^3\)
= \(2^{31}.625^7.5^3^{\left(2\right)}\)
Từ (1) và (2) \(\Rightarrow\) \(2^{31}.625^7.5^3\) > \(2^{31}.512^7.4^3\) > \(2^{31}.312^7.5^3\)
Vậy \(2^{100}< 10^{31}\)