Question Number 6523 by Tawakalitu. last updated on 30/Jun/16
$$\sqrt{{x}^{{x}^{\mathrm{6}} } }\:\:\:=\:\:\:\mathrm{729} \\ $$$$ \\ $$$${find}\:\:{x} \\ $$
Commented by Rasheed Soomro last updated on 01/Jul/16
$$\sqrt{{x}^{{x}^{\mathrm{6}} } }\:\:\:=\:\:\:\mathrm{729} \\ $$$${If}\:\:{x}\:\:{is}\:{a}\:\:{whole}\:{number} \\ $$$${x}^{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{6}} } =\left({x}^{\mathrm{1}/\mathrm{2}} \right)^{{x}^{\mathrm{6}} } =\mathrm{729} \\ $$$${As}\:{lhs}\:{is}\:\:{whole}\:{number} \\ $$$${So}\:{x}\:\:{is}\:\:{perfect}\:\:{square}\:\:\mathrm{OR}\:\:\mathrm{2}\:\mid\:{x}^{\mathrm{6}} \:\:{i}-{e}\:\:{x}\:{is}\:{even} \\ $$$${But}\:{if}\:{x}\:\:{were}\:\:{even}\:{x}^{{x}^{\mathrm{6}} } \:{were}\:{also}\:{even}\:{but} \\ $$$${x}^{{x}^{\mathrm{6}} } \:{is}\:{odd}\:{and}\:{equal}\:{to}\:\mathrm{729}^{\mathrm{2}} \\ $$$${Hence}\:{x}\:{is}\:{an}\:{odd}\:{perfect}\:{square}. \\ $$$${Continue} \\ $$$$ \\ $$
Commented by Temp last updated on 01/Jul/16
$${Do}\:{odd}\:{squares}\:{exist}? \\ $$
Commented by Rasheed Soomro last updated on 01/Jul/16
$${If}\:{x}\:{is}\:{whole}\:{number},\:{then} \\ $$$${x}\:{must}\:{be}\:{odd}\:{square}. \\ $$
Commented by prakash jain last updated on 01/Jul/16
$$\mathrm{729}=\mathrm{3}^{\mathrm{6}} \\ $$$${x}^{{x}^{\mathrm{6}} } =\mathrm{3}^{\mathrm{12}} \\ $$$${try}\:{x}=\mathrm{2} \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{6}} } =\mathrm{2}^{\mathrm{64}} \:\gg\:\mathrm{3}^{\mathrm{12}} \\ $$$$\Rightarrow\mathrm{1}<{x}<\mathrm{2} \\ $$$${y}={x}^{{x}^{\mathrm{6}} } \\ $$$${y}^{'} ={x}^{{x}^{\mathrm{6}} } \centerdot{x}^{\mathrm{5}} \left(\mathrm{ln}\:{x}+\mathrm{1}\right) \\ $$$$\mathrm{Slope}\:\mathrm{of}\:\mathrm{graph}\:\mathrm{is}\:\mathrm{very}\:\mathrm{high}. \\ $$