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:

\[\begin{array}{l} a)\,\,\,{31^{11}} < {17^{14}}\\ b)\,\,{2^{91}} > {5^{35}}\\ c)\,\,{54^4} < \,\,{21^{12}}\\ d)\,\,{333^{444}}\,\, > {444^{333}} \end{array}\]

Giải thích các bước giải: \[\begin{array}{l} a)\,\,{31^{11}},\,\,\,{17^{14}}\\ {31^{11}} < {32^{11}} = {\left( {{2^5}} \right)^{11}} = {2^{55}}\\ {17^{14}} > {16^{14}} = {\left( {{2^4}} \right)^{14}} = {2^{56}}\\ Ma\,\,\,{2^{55}} < {2^{56}}\\ \Rightarrow {31^{11}} < {2^{55}} < {2^{56}} < {17^{14}}\\ \Rightarrow {31^{11}} < {17^{14}}.\\ b)\,\,{2^{91}};\,\,\,{5^{35}}\\ {2^{91}} = {2^{13.7}} = {\left( {{2^{13}}} \right)^7} = {8192^7}\\ {5^{35}} = {5^{5.7}} = {\left( {{5^5}} \right)^7} = {3125^7}\\ Ma\,\,8192 > 3125\\ \Rightarrow {8192^7} > {3125^7}\\ \Rightarrow \,{2^{91}} > {5^{35}}.\\ c)\,\,{54^4}\,;\,\,\,{21^{12}}\\ Ta\,\,\,co:\,\,\,{54^4} = {\left( {27.2} \right)^4} = {27^4}{.2^4} = {\left( {{3^3}} \right)^4}{.2^4} = {3^{12}}{.2^4}\\ {21^{12}} = {\left( {3.7} \right)^{12}} = {3^{12}}{.7^{12}}\\ Vi\,\,\,{2^4} < {7^{12}}\\ \Rightarrow {3^{12}}{.2^4} < {3^{12}}{.7^{12}}\\ \Rightarrow \,{54^4}\, < \,\,{21^{12}}.\\ d)\,\,{333^{444}};\,\,{444^{333}}\\ Ta\,\,\,co:\,\,\,\,{333^{444}} = {\left( {3.111} \right)^{444}} = {\left( {3.111} \right)^{4.111}} = {\left( {{3^4}{{.111}^4}} \right)^{111}}\\ {444^{333}} = {\left( {4.111} \right)^{333}} = {\left( {4.111} \right)^{3.111}} = {\left( {{4^3}{{.111}^3}} \right)^{111}}\\ Co:\,\,\,{3^4}{.111^4} = {111^3}.81.111\\ {4^3}{.111^3} = {111^3}.64\\ Vi\,\,\,{81.111.111^3} > {64.111^3}\\ \Rightarrow {333^{444}} > {444^{333}}. \end{array}\]