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x-x-6-729-find-x-




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

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