Câu hỏi của Lê Thành Đạt

Question

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

Trần Thị Loan · Accepted Answer

$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) => $333^{4^5}$ < $3^{444^5}$ < $3^{4^{555}}$Câu trả lời: $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) => $333^{4^5}$ < $3^{444^5}$ < $3^{4^{555}}$

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