Question Number 167282 by abdurehime last updated on 11/Mar/22
$$\mathrm{solve}\:\left(\mathrm{3}^{\mathrm{x}} \right)^{\mathrm{2}} ×\mathrm{4}^{\mathrm{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$
Answered by nurtani last updated on 11/Mar/22
$$\left(\mathrm{3}^{{x}} \right)^{\mathrm{2}} ×\mathrm{4}^{{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\left(\mathrm{3}^{\mathrm{2}} \right)^{{x}} ×\mathrm{4}^{{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\mathrm{9}^{{x}} ×\mathrm{4}^{{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\left(\mathrm{9}×\mathrm{4}\right)^{{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\mathrm{36}^{{x}} =\sqrt{\mathrm{216}} \\ $$$$\mathrm{6}^{\mathrm{2}{x}} =\mathrm{6}^{\frac{\mathrm{3}}{\mathrm{2}}} \:\Rightarrow\:\mathrm{2}{x}=\frac{\mathrm{3}}{\mathrm{2}}\:\Rightarrow\:{x}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$ \\ $$
Answered by DrHZ last updated on 12/Mar/22
$${x}=? \\ $$$$\left(\mathrm{3}\right)^{\mathrm{2}{x}} ×\mathrm{2}^{\mathrm{2}{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\mathrm{3}^{\mathrm{2}{x}} ×\mathrm{2}^{\mathrm{2}{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\left(\mathrm{3}×\mathrm{2}\right)^{\mathrm{2}{x}} =\sqrt{\mathrm{6}^{\mathrm{2}} ×\mathrm{6}} \\ $$$$\left(\mathrm{6}\right)^{\mathrm{2}{x}} =\sqrt{\mathrm{216}} \\ $$$$\mathrm{6}^{\mathrm{2}{x}} =\mathrm{6}^{\frac{\mathrm{3}}{\mathrm{2}}} \Rightarrow\mathrm{2}{x}=\frac{\mathrm{3}}{\mathrm{2}}\Rightarrow\mathrm{4}{x}=\mathrm{3}\Rightarrow{x}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$−−−−−−−−−−−− \\ $$$$\mathrm{3}^{\mathrm{2}\frac{\mathrm{3}}{\mathrm{4}}} ×\mathrm{2}^{\mathrm{2}\frac{\mathrm{3}}{\mathrm{4}}} =\mathrm{6}\sqrt{\mathrm{6}}\Rightarrow\mathrm{3}^{\frac{\mathrm{3}}{\mathrm{2}}} ×\mathrm{2}^{\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\left(\mathrm{3}×\mathrm{2}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{6}\sqrt{\mathrm{6}}\Rightarrow\left(\mathrm{6}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\sqrt{\mathrm{6}×\mathrm{6}^{\mathrm{2}} =}\mathrm{6}\sqrt{\mathrm{6}} \\ $$$$\mathrm{6}\sqrt{\mathrm{6}}=\mathrm{6}\sqrt{\mathrm{6}} \\ $$$${Dr}\:{HZ} \\ $$