Question Number 7087 by Tawakalitu. last updated on 10/Aug/16
Commented by Yozzii last updated on 10/Aug/16
$${W}=\mathrm{2}^{\mathrm{129}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \\ $$$${X}=\mathrm{2}^{\mathrm{127}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \\ $$$${Y}=\mathrm{2}^{\mathrm{126}} \mathrm{3}^{\mathrm{82}} \mathrm{5}^{\mathrm{131}} \\ $$$${Z}=\mathrm{2}^{\mathrm{125}} \mathrm{3}^{\mathrm{82}} \mathrm{5}^{\mathrm{132}} \\ $$$$−−−−−−−−−−−−−−−−−−−−−−− \\ $$$$\mathrm{2}^{\mathrm{127}} <\mathrm{2}^{\mathrm{129}} \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{127}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} <\mathrm{2}^{\mathrm{129}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \\ $$$${X}<{W} \\ $$$${W}−{Y}=\mathrm{2}^{\mathrm{126}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \left(\mathrm{2}^{\mathrm{3}} −\mathrm{3}\right)>\mathrm{0}\Rightarrow{W}>{Y} \\ $$$${X}−{Y}=\mathrm{2}^{\mathrm{126}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \left(\mathrm{2}−\mathrm{3}\right)<\mathrm{0}\Rightarrow{Y}>{X} \\ $$$$\therefore{W}>{Y}>{X} \\ $$$${W}−{Z}=\mathrm{2}^{\mathrm{125}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \left(\mathrm{2}^{\mathrm{4}} −\mathrm{3}×\mathrm{5}\right)>\mathrm{0}\Rightarrow{W}>{Z} \\ $$$${Z}−{Y}=\mathrm{2}^{\mathrm{125}} \mathrm{3}^{\mathrm{82}} \mathrm{5}^{\mathrm{131}} \left(\mathrm{5}−\mathrm{2}\right)>\mathrm{0}\Rightarrow{Z}>{Y}>{X} \\ $$$$ \\ $$$$\therefore\:{X}<{Y}<{Z}<{W} \\ $$
Commented by Tawakalitu. last updated on 10/Aug/16
$${Great},\:{i}\:{appreciate}\:..\:{thanks}\:{so}\:{much}. \\ $$
Commented by Rasheed Soomro last updated on 10/Aug/16
$$\underset{−} {{Simple}\:{approach}} \\ $$$${W}=\mathrm{2}^{\mathrm{129}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \\ $$$${X}=\mathrm{2}^{\mathrm{127}} \mathrm{3}^{\mathrm{81}} \mathrm{5}^{\mathrm{131}} \\ $$$${Y}=\mathrm{2}^{\mathrm{126}} \mathrm{3}^{\mathrm{82}} \mathrm{5}^{\mathrm{131}} \\ $$$${Z}=\mathrm{2}^{\mathrm{125}} \mathrm{3}^{\mathrm{82}} \mathrm{5}^{\mathrm{132}} \\ $$$$−−−−−−−−−−−−−−−−− \\ $$$${Dividing}\:{by}\:{same}\:{positive}\:{number} \\ $$$${will}\:{not}\:{affect}\:{inequality}/{equality}. \\ $$$${So}\:{dividing}\:{by}\:\mathrm{2}^{\mathrm{125}} ×\mathrm{3}^{\mathrm{81}} ×\mathrm{5}^{\mathrm{131}} \:\left({GCD}\right)\:{to}\:{W}, \\ $$$${X},{Y}\:{and}\:{Z}\:{to}\:{obtain}\:{new}\:{quantities} \\ $$$${W}\:',{X}^{\:'} ,{Y}^{\:'} \:{and}\:{Z}\:^{'} \:{respectively}. \\ $$$${W}\:'=\mathrm{2}^{\mathrm{4}} =\mathrm{16} \\ $$$${X}'=\mathrm{2}^{\mathrm{2}} =\mathrm{4} \\ $$$${Y}\:'=\mathrm{2}×\mathrm{3}=\mathrm{6} \\ $$$${Z}\:'=\mathrm{3}×\mathrm{5}=\mathrm{15} \\ $$$${Clearly}\:\mathrm{4}<\mathrm{6}<\mathrm{15}<\mathrm{16} \\ $$$${So},\:\:\:\:\:\:\:\:{X}'<{Y}\:'<{Z}\:'<{W}\:' \\ $$$${Or}\:\:\:\:\:\:\:\:\:\:\:{X}<{Y}<{Z}<{W} \\ $$
Commented by Tawakalitu. last updated on 10/Aug/16
$${Wow}\:..\:{i}\:{really}\:{appreciate}\:{your}\:{effort}...\:{Thanks}\:{so}\:{much} \\ $$