Menu Close

3-12-1-3-3-1-3-x-1-3-y-1-3-z-1-3-x-y-z-N-x-y-z-please-help-me-

Tinku Tara 0 Comments
Pin




Question Number 195855 by jabarsing last updated on 11/Aug/23
{ ((3(√(((12))^(1/3) −(3)^(1/3) ))=(x)^(1/3) +(y)^(1/3) −(z)^(1/3) )),((x,y,z ∈ N)) :} ⇒ x,y,z =? please help me
$$\begin{cases}{\mathrm{3}\sqrt{\sqrt[{\mathrm{3}}]{\mathrm{12}}−\sqrt[{\mathrm{3}}]{\mathrm{3}}}=\sqrt[{\mathrm{3}}]{{x}}+\sqrt[{\mathrm{3}}]{{y}}−\sqrt[{\mathrm{3}}]{{z}}}\\{{x},{y},{z}\:\in\:{N}}\end{cases}\:\:\Rightarrow\:{x},{y},{z}\:=? \\ $$$${please}\:{help}\:{me} \\ $$
Answered by Rasheed.Sindhi last updated on 13/Aug/23
Unsuccessful Try... { ((3(√(((12))^(1/3) −(3)^(1/3) ))=(x)^(1/3) +(y)^(1/3) −(z)^(1/3) )),((x,y,z ∈ N)) :} ⇒ x,y,z =? (3(√(((12))^(1/3) −(3)^(1/3) )))^3 =((x)^(1/3) +(y)^(1/3) −(z)^(1/3) )^3 (a+b−c)^3 =a^3 +b^3 −c^3 +3ab(a+b)−3bc(b−c)−3ac(a−c)−6abc 27(√((((12))^(1/3) −(3)^(1/3) )^3 )) =x+y−z+3((xy))^(1/3) ((x)^(1/3) +(y)^(1/3) ) −3((yz))^(1/3) ((y)^(1/3) −(z)^(1/3) ) −3((xz))^(1/3) ((x)^(1/3) −(z)^(1/3) ) −6((xyz))^(1/3) (a−b)^3 =a^3 −b^3 −3ab(a−b) LHS: 27(√(12−3−3((36))^(1/3) (((12))^(1/3) −(3)^(1/3) ))) =27(√(9−((36))^(1/3) (((12))^(1/3) −(3)^(1/3) ))) =27(√(9−((36))^(1/3) (3)^(1/3) ((4)^(1/3) −1 ))) =27(√(9−3(4)^(1/3) ((4)^(1/3) −1 ))) =27(√(9−6(2)^(1/3) −3(4)^(1/3) )) (LHS)^2 =729(9−6(2)^(1/3) −3(4)^(1/3) ) RHS: =x+y−z+3((xy))^(1/3) ((x)^(1/3) +(y)^(1/3) ) −3((yz))^(1/3) ((y)^(1/3) −(z)^(1/3) ) −3((xz))^(1/3) ((x)^(1/3) −(z)^(1/3) ) −6((xyz))^(1/3) (RHS)^2 =(x+y−z+3((xy))^(1/3) ((x)^(1/3) +(y)^(1/3) ) −3((yz))^(1/3) ((y)^(1/3) −(z)^(1/3) ) −3((xz))^(1/3) ((x)^(1/3) −(z)^(1/3) ) −6((xyz))^(1/3) )^2 (LHS)^2 =(RHS)^2 Too complicated .....
$$\mathrm{Unsuccessful}\:\mathrm{Try}… \\ $$$$\begin{cases}{\mathrm{3}\sqrt{\sqrt[{\mathrm{3}}]{\mathrm{12}}−\sqrt[{\mathrm{3}}]{\mathrm{3}}}=\sqrt[{\mathrm{3}}]{{x}}+\sqrt[{\mathrm{3}}]{{y}}−\sqrt[{\mathrm{3}}]{{z}}}\\{{x},{y},{z}\:\in\:{N}}\end{cases}\:\:\Rightarrow\:{x},{y},{z}\:=? \\ $$$$\left(\mathrm{3}\sqrt{\sqrt[{\mathrm{3}}]{\mathrm{12}}−\sqrt[{\mathrm{3}}]{\mathrm{3}}}\right)^{\mathrm{3}} =\left(\sqrt[{\mathrm{3}}]{{x}}+\sqrt[{\mathrm{3}}]{{y}}−\sqrt[{\mathrm{3}}]{{z}}\right)^{\mathrm{3}} \\ $$$$\left({a}+{b}−{c}\right)^{\mathrm{3}} \\ $$$$={a}^{\mathrm{3}} +{b}^{\mathrm{3}} −{c}^{\mathrm{3}} +\mathrm{3}{ab}\left({a}+{b}\right)−\mathrm{3}{bc}\left({b}−{c}\right)−\mathrm{3}{ac}\left({a}−{c}\right)−\mathrm{6}{abc} \\ $$$$\mathrm{27}\sqrt{\left(\sqrt[{\mathrm{3}}]{\mathrm{12}}\:−\sqrt[{\mathrm{3}}]{\mathrm{3}}\:\:\right)^{\mathrm{3}} }\: \\ $$$$\:\:\:\:\:\:\:\:\:={x}+{y}−{z}+\mathrm{3}\sqrt[{\mathrm{3}}]{{xy}}\:\left(\sqrt[{\mathrm{3}}]{{x}}\:+\sqrt[{\mathrm{3}}]{{y}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{{yz}}\:\left(\sqrt[{\mathrm{3}}]{{y}}\:−\sqrt[{\mathrm{3}}]{{z}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{{xz}}\:\left(\sqrt[{\mathrm{3}}]{{x}}\:−\sqrt[{\mathrm{3}}]{{z}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{6}\sqrt[{\mathrm{3}}]{{xyz}}\: \\ $$$$\left({a}−{b}\right)^{\mathrm{3}} ={a}^{\mathrm{3}} −{b}^{\mathrm{3}} −\mathrm{3}{ab}\left({a}−{b}\right) \\ $$$${LHS}: \\ $$$$\mathrm{27}\sqrt{\mathrm{12}−\mathrm{3}−\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{36}}\:\left(\sqrt[{\mathrm{3}}]{\mathrm{12}}\:−\sqrt[{\mathrm{3}}]{\mathrm{3}}\right)}\: \\ $$$$=\mathrm{27}\sqrt{\mathrm{9}−\sqrt[{\mathrm{3}}]{\mathrm{36}}\:\left(\sqrt[{\mathrm{3}}]{\mathrm{12}}\:−\sqrt[{\mathrm{3}}]{\mathrm{3}}\:\right)} \\ $$$$=\mathrm{27}\sqrt{\mathrm{9}−\sqrt[{\mathrm{3}}]{\mathrm{36}}\:\sqrt[{\mathrm{3}}]{\mathrm{3}}\left(\sqrt[{\mathrm{3}}]{\mathrm{4}}\:−\mathrm{1}\:\right)}\: \\ $$$$=\mathrm{27}\sqrt{\mathrm{9}−\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{4}}\:\left(\sqrt[{\mathrm{3}}]{\mathrm{4}}\:−\mathrm{1}\:\right)}\: \\ $$$$=\mathrm{27}\sqrt{\mathrm{9}−\mathrm{6}\sqrt[{\mathrm{3}}]{\mathrm{2}}\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{4}}\:}\: \\ $$$$\left({LHS}\right)^{\mathrm{2}} =\mathrm{729}\left(\mathrm{9}−\mathrm{6}\sqrt[{\mathrm{3}}]{\mathrm{2}}\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{4}}\:\right) \\ $$$${RHS}: \\ $$$$\:\:\:\:\:\:\:\:\:={x}+{y}−{z}+\mathrm{3}\sqrt[{\mathrm{3}}]{{xy}}\:\left(\sqrt[{\mathrm{3}}]{{x}}\:+\sqrt[{\mathrm{3}}]{{y}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{{yz}}\:\left(\sqrt[{\mathrm{3}}]{{y}}\:−\sqrt[{\mathrm{3}}]{{z}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{{xz}}\:\left(\sqrt[{\mathrm{3}}]{{x}}\:−\sqrt[{\mathrm{3}}]{{z}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{6}\sqrt[{\mathrm{3}}]{{xyz}}\: \\ $$$$\left({RHS}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:=\left({x}+{y}−{z}+\mathrm{3}\sqrt[{\mathrm{3}}]{{xy}}\:\left(\sqrt[{\mathrm{3}}]{{x}}\:+\sqrt[{\mathrm{3}}]{{y}}\:\right)\right. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{{yz}}\:\left(\sqrt[{\mathrm{3}}]{{y}}\:−\sqrt[{\mathrm{3}}]{{z}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{3}\sqrt[{\mathrm{3}}]{{xz}}\:\left(\sqrt[{\mathrm{3}}]{{x}}\:−\sqrt[{\mathrm{3}}]{{z}}\:\right) \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{6}\sqrt[{\mathrm{3}}]{{xyz}}\:\right)^{\mathrm{2}} \\ $$$$\: \\ $$$$\left({LHS}\right)^{\mathrm{2}} =\left({RHS}\right)^{\mathrm{2}} \\ $$$${Too}\:{complicated} \\ $$$$….. \\ $$$$\: \\ $$

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *