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Question Number 165984 by nurtani last updated on 10/Feb/22

Commented by cortano1 last updated on 10/Feb/22

2024

$$\mathrm{2024} \\ $$

Answered by Rasheed.Sindhi last updated on 11/Feb/22

{ ((x^2 +y^2 +z^2 =70........(i))),((x^3 +y^3 +z^3 =64........(ii))),(((x+y)(y+z)(z+x)=−24....(iii))),((x^4 +y^4 +z^4 +2=?)) :} •(x+y+z)^3 =x^3 +y^3 +z^3 +3(x+y)(y+z)(z+x) (x+y+z)^3 =64+3(−24)=−8 determinant ((( x+y+z=−2))) • (x+y+z)^2 =(−2)^2 x^2 +y^2 +z^2 +2(xy+yz+zx)=4 70+2(xy+yz+zx)=4 determinant (((xy+yz+zx=−33))) •x^3 +y^3 +z^3 −3xyz=(x+y+z)(x^2 +y^2 +z^2 −xy−yz−zx) 64−3xyz=(−2)(70−(−33)) xyz=((64+206)/3)=((270)/3)=90 determinant (((xyz=90))) •(xy+yz+zx)^2 =(−33)^2 x^2 y^2 +y^2 z^2 +z^2 x^2 +2xyz(x+y+z)=1089 x^2 y^2 +y^2 z^2 +z^2 x^2 +2(90)(−2)=1089 x^2 y^2 +y^2 z^2 +z^2 x^2 =1089+360=1449 determinant (((x^2 y^2 +y^2 z^2 +z^2 x^2 =1449))) •(x^2 +y^2 +z^2 )^2 =x^4 +y^4 +z^4 +2(x^2 y^2 +y^2 z^2 +z^2 x^2 ) (70)^2 =x^4 +y^4 +z^4 +2(1449) x^4 +y^4 +z^4 =2002 x^4 +y^4 +z^4 +2=2004 determinant (((x^4 +y^4 +z^4 +2=2004)))

$$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{70}........\left({i}\right)}\\{{x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =\mathrm{64}........\left({ii}\right)}\\{\left({x}+{y}\right)\left({y}+{z}\right)\left({z}+{x}\right)=−\mathrm{24}....\left({iii}\right)}\\{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} +\mathrm{2}=?}\end{cases} \\ $$$$\bullet\left({x}+{y}+{z}\right)^{\mathrm{3}} ={x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} +\mathrm{3}\left({x}+{y}\right)\left({y}+{z}\right)\left({z}+{x}\right) \\ $$$$\left({x}+{y}+{z}\right)^{\mathrm{3}} =\mathrm{64}+\mathrm{3}\left(−\mathrm{24}\right)=−\mathrm{8} \\ $$$$\begin{array}{|c|}{\:{x}+{y}+{z}=−\mathrm{2}}\\\hline\end{array} \\ $$$$\bullet\:\left({x}+{y}+{z}\right)^{\mathrm{2}} =\left(−\mathrm{2}\right)^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +\mathrm{2}\left({xy}+{yz}+{zx}\right)=\mathrm{4} \\ $$$$\mathrm{70}+\mathrm{2}\left({xy}+{yz}+{zx}\right)=\mathrm{4} \\ $$$$\begin{array}{|c|}{{xy}+{yz}+{zx}=−\mathrm{33}}\\\hline\end{array} \\ $$$$\bullet{x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} −\mathrm{3}{xyz}=\left({x}+{y}+{z}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} −{xy}−{yz}−{zx}\right) \\ $$$$\mathrm{64}−\mathrm{3}{xyz}=\left(−\mathrm{2}\right)\left(\mathrm{70}−\left(−\mathrm{33}\right)\right) \\ $$$${xyz}=\frac{\mathrm{64}+\mathrm{206}}{\mathrm{3}}=\frac{\mathrm{270}}{\mathrm{3}}=\mathrm{90} \\ $$$$\begin{array}{|c|}{{xyz}=\mathrm{90}}\\\hline\end{array} \\ $$$$\bullet\left({xy}+{yz}+{zx}\right)^{\mathrm{2}} =\left(−\mathrm{33}\right)^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{2}} {z}^{\mathrm{2}} +{z}^{\mathrm{2}} {x}^{\mathrm{2}} +\mathrm{2}{xyz}\left({x}+{y}+{z}\right)=\mathrm{1089} \\ $$$${x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{2}} {z}^{\mathrm{2}} +{z}^{\mathrm{2}} {x}^{\mathrm{2}} +\mathrm{2}\left(\mathrm{90}\right)\left(−\mathrm{2}\right)=\mathrm{1089} \\ $$$${x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{2}} {z}^{\mathrm{2}} +{z}^{\mathrm{2}} {x}^{\mathrm{2}} =\mathrm{1089}+\mathrm{360}=\mathrm{1449} \\ $$$$\begin{array}{|c|}{{x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{2}} {z}^{\mathrm{2}} +{z}^{\mathrm{2}} {x}^{\mathrm{2}} =\mathrm{1449}}\\\hline\end{array} \\ $$$$\bullet\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)^{\mathrm{2}} ={x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} +\mathrm{2}\left({x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{2}} {z}^{\mathrm{2}} +{z}^{\mathrm{2}} {x}^{\mathrm{2}} \right) \\ $$$$\left(\mathrm{70}\right)^{\mathrm{2}} ={x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} +\mathrm{2}\left(\mathrm{1449}\right) \\ $$$${x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} =\mathrm{2002} \\ $$$${x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} +\mathrm{2}=\mathrm{2004} \\ $$$$\begin{array}{|c|}{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} +\mathrm{2}=\mathrm{2004}}\\\hline\end{array} \\ $$

Commented by Rasheed.Sindhi last updated on 11/Feb/22

Thanks for rapid feedback sir! (Here some persons even don′t give any feedback.)

$$\mathbb{T}\mathrm{han}\Bbbk\mathrm{s}\:\mathrm{for}\:\mathrm{rapid}\:\mathrm{feedback}\:\mathrm{sir}! \\ $$$$\left(\mathrm{Here}\:\mathrm{some}\:\mathrm{persons}\:\mathrm{even}\:\mathrm{don}'\mathrm{t}\:\mathrm{give}\right. \\ $$$$\left.\mathrm{any}\:\mathrm{feedback}.\right) \\ $$

Commented by nurtani last updated on 11/Feb/22

Nice, Thank you sir

$${Nice},\:{Thank}\:{you}\:{sir} \\ $$

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