Question

[Lê Thành Đạt]

Hãy xếp \(333^{4^5},3^{444^5},3^{4^{555}}\)theo thứ tự tăng dần.

Answer 1

[Trần Thị Loan]

\(333^{4^5}=\left(3.111\right)^{4^5}=3^{4^5}.111^{4^5}\)

\(3^{444^5}=3^{\left(4.111\right)^5}=3^{4^5.111^5}=\left(3^{4^5}\right)^{111^5}=3^{4^5}.\left(3^{4^5}\right)^{111^5-1}=3^{4^5}.\left(81^5\right)^{111^5-1}\)

\(3^{4^{555}}=3^{4^5.4^{550}}=\left(3^{4^5}\right)^{4^{550}}\)

+) Dễ có: \(3^{4^5}.111^{4^5}\) < \(3^{4^5}.\left(81^5\right)^{111^5-1}\)

=> \(333^{4^5}\)  < \(3^{444^5}\)   (1)

+) Ta có: \(\left(3^{4^5}\right)^{111^5}\) < \(\left(3^{4^5}\right)^{4^{550}}\) vì \(111^5\) < \(4^{550}=\left(4^5\right)^{110}=1024^{110}\)

=> \(3^{444^5}\) < \(3^{4^{555}}\) (2)

(1)(2) =>  \(333^{4^5}\)  < \(3^{444^5}\) < \(3^{4^{555}}\)