Question Number 74840 by aliesam last updated on 01/Dec/19
Commented by kaivan.ahmadi last updated on 01/Dec/19
$${a}^{\mathrm{4}} =\mathrm{1}\Rightarrow \\ $$$${b}^{\mathrm{4}} ={a}^{\mathrm{4}} =\mathrm{1}\Rightarrow{o}\left({b}\right)=\mathrm{4} \\ $$$$\left({a}^{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{1}\Rightarrow{o}\left({a}^{\mathrm{2}} \right)=\mathrm{2} \\ $$$${o}\left({a}^{\mathrm{3}} \right)={o}\left({a}.{a}^{\mathrm{2}} \right)=\frac{{o}\left({a}\right){o}\left({a}^{\mathrm{2}} \right)}{\left({o}\left({a}\right),{o}\left({a}^{\mathrm{2}} \right)\right)}=\frac{\mathrm{2}×\mathrm{4}}{\mathrm{2}}=\mathrm{4} \\ $$$${o}\left({ab}\right)=\frac{{o}\left({a}\right){o}\left({b}\right)}{\left({o}\left({a}\right),{o}\left({b}\right)\right)}=\frac{\mathrm{4}×\mathrm{4}}{\mathrm{4}}=\mathrm{4} \\ $$$${o}\left({a}^{\mathrm{2}} {b}\right)=\frac{\mathrm{2}×\mathrm{4}}{\mathrm{2}}=\mathrm{4} \\ $$$${o}\left({a}^{\mathrm{3}} {b}\right)=\frac{\mathrm{4}×\mathrm{4}}{\mathrm{4}}=\mathrm{4} \\ $$
Commented by kaivan.ahmadi last updated on 01/Dec/19
$${a}.{a}^{\mathrm{3}} =\mathrm{1}\Rightarrow{a}^{−\mathrm{1}} ={a}^{\mathrm{3}} \:,\left({a}^{\mathrm{3}} \right)^{−\mathrm{1}} ={a} \\ $$$$\left({a}^{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{1}\Rightarrow\left({a}^{\mathrm{2}} \right)^{−\mathrm{1}} ={a}^{\mathrm{2}} \\ $$$$\left({a}^{\mathrm{2}} {b}\right){b}={a}^{\mathrm{2}} {b}^{\mathrm{2}} ={b}^{\mathrm{2}} {b}^{\mathrm{2}} ={b}^{\mathrm{4}} =\mathrm{1}\Rightarrow\left({a}^{\mathrm{2}} {b}\right)^{−\mathrm{1}} ={b},{b}^{−\mathrm{1}} ={a}^{\mathrm{2}} {b} \\ $$$$ \\ $$
Commented by kaivan.ahmadi last updated on 02/Dec/19
