Question Number 227369 by Math1 last updated on 18/Jan/26
$$\begin{cases}{\mathrm{3}^{\boldsymbol{\mathrm{m}}} \:+\:\mathrm{2}\centerdot\mathrm{4}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:=\:\mathrm{17}}\\{\mathrm{4}^{\boldsymbol{\mathrm{n}}} \:−\:\mathrm{5}\centerdot\mathrm{3}^{\boldsymbol{\mathrm{m}}} \:=\:−\mathrm{44}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\mathrm{m}+\mathrm{n}=? \\ $$
Answered by A5T last updated on 19/Jan/26
$$\mathrm{5}×\left(\mathrm{i}\right)−\left(\mathrm{ii}\right)\:\Rightarrow\:\mathrm{5}\centerdot\mathrm{2}\centerdot\mathrm{4}\centerdot\mathrm{4}^{\mathrm{n}} +\mathrm{4}^{\mathrm{n}} =\mathrm{5}×\mathrm{17}−\mathrm{44} \\ $$$$\Rightarrow\mathrm{41}\centerdot\mathrm{4}^{\mathrm{n}} =\mathrm{41} \\ $$$$\:\Rightarrow\mathrm{n}=\mathrm{0}\:\Rightarrow\mathrm{m}=\mathrm{2} \\ $$