Question Number 82489 by niroj last updated on 21/Feb/20
$$ \\ $$$$\: \\ $$$$\mathrm{If}\:\mathrm{a}^{\mathrm{b}} =\mathrm{b}^{\mathrm{a}} \:\mathrm{and}\:\mathrm{a}=\mathrm{2b}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\ $$$$\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \:. \\ $$
Answered by mind is power last updated on 21/Feb/20
$$\Rightarrow\left(\mathrm{2}{b}\right)^{{b}} ={b}^{\mathrm{2}{b}} \\ $$$$\Rightarrow{b}^{{b}} \left(\mathrm{2}^{{b}} −{b}^{{b}} \right)=\mathrm{0} \\ $$$${since}\:{b}\neq\mathrm{0}\:{cause}\Rightarrow{a}=\mathrm{0}\:{and}\:\mathrm{0}^{\mathrm{0}} \:{not}\:{defind} \\ $$$$\Rightarrow\mathrm{2}^{{b}} −{b}^{{b}} =\mathrm{0}\Rightarrow{bln}\left(\mathrm{2}\right)={bln}\left({b}\right)\Rightarrow{b}\left({ln}\left(\mathrm{2}\right)−{ln}\left({b}\right)\right)=\mathrm{0} \\ $$$$\Rightarrow{b}=\mathrm{2}\Rightarrow{a}=\mathrm{4} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{4}+\mathrm{16}=\mathrm{20} \\ $$
Commented by niroj last updated on 21/Feb/20
$$\:{thank}\:{you}. \\ $$