So sánh: a) 31^11 và 17^14 b) 2^91 và 5^35 c) 54^4 và 21^12 d) 333^444 và 444^333

Đáp án:

\(\eqalign{
& a)\,\,{31^{11}} < {17^{14}} \cr & b)\,\,{5^{35}} < {2^{91}} \cr & c)\,\,{54^4} < {21^{12}} \cr & d)\,\,{444^{333}} < {333^{444}} \cr} \)

Giải thích các bước giải:

$$\eqalign{
& a)\,\,{31^{11}} < {32^{11}} = {\left( {{2^5}} \right)^{11}} = {2^{55}} \cr & \,\,\,\,\,\,\,{17^{14}} > {16^{14}} = {\left( {{2^4}} \right)^{14}} = {2^{56}} \cr
& \Rightarrow {31^{11}} < {2^{55}} < {2^{56}} < {17^{14}} \cr & Vay\,\,{31^{11}} < {17^{14}} \cr & b)\,\,{2^{91}} = {2^{13.7}} = {\left( {{2^{13}}} \right)^7} = {8192^7} \cr & \,\,\,\,\,\,{5^{35}} = {5^{5.7}} = {\left( {{5^5}} \right)^7} = {3125^7} \cr & Ta\,\,co:\,\,{3125^7} < {8192^7} \cr & Vay\,\,{5^{35}} < {2^{91}} \cr & c)\,\,{54^4} \cr & \,\,\,\,\,\,{21^{12}} = {21^{3.4}} = {\left( {{{21}^3}} \right)^4} = {9261^4} \cr & Ta\,\,co:\,\,{54^4} < {9261^4} \cr & Vay\,\,{54^4} < {21^{12}} \cr & d)\,\,{333^{444}} = {333^{4.111}} = {\left( {{{333}^4}} \right)^{111}} = {\left( {{{111}^4}{{.3}^4}} \right)^{111}} = {\left( {{{111}^3}.8991} \right)^{111}} \cr & \,\,\,\,\,\,\,{444^{333}} = {444^{3.111}} = {\left( {{{444}^3}} \right)^{111}} = {\left( {{{111}^3}{{.4}^3}} \right)^{111}} = {\left( {{{111}^3}.64} \right)^{111}} \cr & Ta\,\,co:\,\,64 < 8991 \cr & \Rightarrow {111^3}.64 < {111^3}.8991 \cr & \Rightarrow {\left( {{{111}^3}.64} \right)^{111}} < {\left( {{{111}^3}.8991} \right)^{111}} \cr & Vay\,\,{444^{333}} < {333^{444}} \cr} $$