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
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:
\[\begin{array}{l}
a)\,\,{31^{11}},\,\,\,{17^{14}}\\
{31^{11}} < {32^{11}} = {\left( {{2^5}} \right)^{11}} = {2^{55}}\\ {17^{14}} > {16^{14}} = {\left( {{2^4}} \right)^{14}} = {2^{56}}\\
Ma\,\,\,{2^{55}} < {2^{56}}\\ \Rightarrow {31^{11}} < {2^{55}} < {2^{56}} < {17^{14}}\\ \Rightarrow {31^{11}} < {17^{14}}.\\ b)\,\,{2^{91}};\,\,\,{5^{35}}\\ {2^{91}} = {2^{13.7}} = {\left( {{2^{13}}} \right)^7} = {8192^7}\\ {5^{35}} = {5^{5.7}} = {\left( {{5^5}} \right)^7} = {3125^7}\\ Ma\,\,8192 > 3125\\
\Rightarrow {8192^7} > {3125^7}\\
\Rightarrow \,{2^{91}} > {5^{35}}.\\
c)\,\,{54^4}\,;\,\,\,{21^{12}}\\
Ta\,\,\,co:\,\,\,{54^4} = {\left( {27.2} \right)^4} = {27^4}{.2^4} = {\left( {{3^3}} \right)^4}{.2^4} = {3^{12}}{.2^4}\\
{21^{12}} = {\left( {3.7} \right)^{12}} = {3^{12}}{.7^{12}}\\
Vi\,\,\,{2^4} < {7^{12}}\\ \Rightarrow {3^{12}}{.2^4} < {3^{12}}{.7^{12}}\\ \Rightarrow \,{54^4}\, < \,\,{21^{12}}.\\ d)\,\,{333^{444}};\,\,{444^{333}}\\ Ta\,\,\,co:\,\,\,\,{333^{444}} = {\left( {3.111} \right)^{444}} = {\left( {3.111} \right)^{4.111}} = {\left( {{3^4}{{.111}^4}} \right)^{111}}\\ {444^{333}} = {\left( {4.111} \right)^{333}} = {\left( {4.111} \right)^{3.111}} = {\left( {{4^3}{{.111}^3}} \right)^{111}}\\ Co:\,\,\,{3^4}{.111^4} = {111^3}.81.111\\ {4^3}{.111^3} = {111^3}.64\\ Vi\,\,\,{81.111.111^3} > {64.111^3}\\
\Rightarrow {333^{444}} > {444^{333}}.
\end{array}\]
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:
$$\eqalign{
& a)\,\,{31^{11}} < {32^{11}} = {\left( {{2^5}} \right)^{11}} = {2^{55}} \cr & \,\,\,\,\,\,\,{17^{14}} > {16^{14}} = {\left( {{2^4}} \right)^{14}} = {2^{56}} \cr
& \Rightarrow {31^{11}} < {2^{55}} < {2^{56}} < {17^{14}} \cr & Vay\,\,{31^{11}} < {17^{14}} \cr & b)\,\,{2^{91}} = {2^{13.7}} = {\left( {{2^{13}}} \right)^7} = {8192^7} \cr & \,\,\,\,\,\,{5^{35}} = {5^{5.7}} = {\left( {{5^5}} \right)^7} = {3125^7} \cr & Ta\,\,co:\,\,{3125^7} < {8192^7} \cr & Vay\,\,{5^{35}} < {2^{91}} \cr & c)\,\,{54^4} \cr & \,\,\,\,\,\,{21^{12}} = {21^{3.4}} = {\left( {{{21}^3}} \right)^4} = {9261^4} \cr & Ta\,\,co:\,\,{54^4} < {9261^4} \cr & Vay\,\,{54^4} < {21^{12}} \cr & d)\,\,{333^{444}} = {333^{4.111}} = {\left( {{{333}^4}} \right)^{111}} = {\left( {{{111}^4}{{.3}^4}} \right)^{111}} = {\left( {{{111}^3}.8991} \right)^{111}} \cr & \,\,\,\,\,\,\,{444^{333}} = {444^{3.111}} = {\left( {{{444}^3}} \right)^{111}} = {\left( {{{111}^3}{{.4}^3}} \right)^{111}} = {\left( {{{111}^3}.64} \right)^{111}} \cr & Ta\,\,co:\,\,64 < 8991 \cr & \Rightarrow {111^3}.64 < {111^3}.8991 \cr & \Rightarrow {\left( {{{111}^3}.64} \right)^{111}} < {\left( {{{111}^3}.8991} \right)^{111}} \cr & Vay\,\,{444^{333}} < {333^{444}} \cr} $$