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