Question

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Answer 1

Respuesta:

E = - 6

Explicación paso a paso:

Answer 2

Respuesta:

E = -6

Explicación paso a paso:

Sabemos:

$x= \sqrt{5}-\sqrt{3} \\y = \sqrt{2}-\sqrt{5}\\z = \sqrt{3}-\sqrt{2}$

Calcular los cuadrados:

$x^2 = (\sqrt{5} -\sqrt{3} )^2 = 5 - 2\sqrt{5}\sqrt{3} + 3 = 8 - 2\sqrt{15} \\y^2 = (\sqrt{2} -\sqrt{5} )^2 = 2 - 2\sqrt{2}\sqrt{5} + 5 = 7 - 2\sqrt{10}\\z^2 = (\sqrt{3} -\sqrt{2} )^2 = 3 - 2\sqrt{3}\sqrt{2} + 2 = 5 - 2\sqrt{6}$

Calcular productos:

$xy = (\sqrt{5}-\sqrt{3})(\sqrt{2}-\sqrt{5}) = \sqrt{10}-5-\sqrt{6}+\sqrt{15} \\yz = (\sqrt{2}-\sqrt{5})(\sqrt{3}-\sqrt{2}) = \sqrt{6}-2-\sqrt{15}+\sqrt{10} \\zx = (\sqrt{3}-\sqrt{2})(\sqrt{5}-\sqrt{3}) = \sqrt{15}-3-\sqrt{10}+\sqrt{6}$

Calcular suma $x^2+y^2+z^2$:

$x^2+y^2+z^2 = 8 - 2\sqrt{15} + 7 - 2\sqrt{10} + 5 -2\sqrt{6} = \\\\ -2\sqrt{15}-2\sqrt{10}-2\sqrt{6}+8+7+5 = \\\\ -2\sqrt{15}-2\sqrt{10}-2\sqrt{6} + 20 =\\\\-2(\sqrt{15}+\sqrt{10}+\sqrt{6}-10)$

Calcular suma $xy+yz+xz$:

$\sqrt{10}-5-\sqrt{6}+\sqrt{15}+\sqrt{6}-2-\sqrt{15}+\sqrt{10}+\sqrt{15}-3-\sqrt{10}+\sqrt{6}=\\\\\sqrt{15}+\sqrt{10}+\sqrt{6}-10$

Calcular la suma $\frac{x^2}{yz} + \frac{y^2}{xz}+\frac{z^2}{xy}$:

$\frac{x^2x^2yz+y^2y^2xz+z^2z^2xy}{ x^2y^2z^2 }$

Calcular $x^2x^2yz$:

$(8-2\sqrt{15})(8-2\sqrt{15})(\sqrt{6}-2-\sqrt{15} + \sqrt{10}) =\\\\(64-2(8)(2\sqrt{15})+(2\sqrt{15})^2)(\sqrt{6}-2-\sqrt{15} + \sqrt{10})=\\\\(64-32\sqrt{15}+60)(\sqrt{6}-2-\sqrt{15} + \sqrt{10})=\\\\4(31-8\sqrt{15})(\sqrt{6}-2-\sqrt{15} + \sqrt{10})=\\\\4(31\sqrt{6}-62-31\sqrt{15}+31\sqrt{10}-8\sqrt{90}+16\sqrt{15}+120-8\sqrt{150})=\\\\4(31\sqrt{6}-62-31\sqrt{15}+31\sqrt{10}-24\sqrt{10}+16\sqrt{15}+120-40\sqrt{6})=\\\\4(-15\sqrt{15}+7\sqrt{10}-9\sqrt{6}+58)=\\\\-60\sqrt{15}+28\sqrt{10}-36\sqrt{6}+232$

Calcular $y^2y^2xz$

$(7-2\sqrt{10})(7-2\sqrt{10})(\sqrt{15}-3-\sqrt{10}+\sqrt{6})=\\\\(49-2(7)(2\sqrt{10})+(2\sqrt{10})^2)(\sqrt{15}-3-\sqrt{10}+\sqrt{6})=\\\\(49-28\sqrt{10}+40)(\sqrt{15}-3-\sqrt{10}+\sqrt{6})=\\\\(89-28\sqrt{10})(\sqrt{15}-3-\sqrt{10}+\sqrt{6})=\\\\89\sqrt{15}-267-89\sqrt{10}+89\sqrt{6}-28\sqrt{150}+84\sqrt{10}+280-28\sqrt{60}=\\\\89\sqrt{15}-267-89\sqrt{10}+89\sqrt{6}-140\sqrt{6}+84\sqrt{10}+280-56\sqrt{15}=\\\\33\sqrt{15}-5\sqrt{10}-51\sqrt{6}+13$

Calcular $z^2z^2xy$:

$(5-2\sqrt{6})(5-2\sqrt{6})(\sqrt{10}-5-\sqrt{6}+\sqrt{15})=\\\\(49-20\sqrt{6})(\sqrt{10}-5-\sqrt{6}+\sqrt{15})=\\\\49\sqrt{10}-245-49\sqrt{6}+49\sqrt{15}-20\sqrt{60}+100\sqrt{6}+120-20\sqrt{90}=\\\\49\sqrt{10}-245-49\sqrt{6}+49\sqrt{15}-40\sqrt{15}+100\sqrt{6}+120-60\sqrt{10}=\\\\-11\sqrt{10}+9\sqrt{15}+51\sqrt{6}-125$

Calcular $x^2y^2z^2$:

$(8-2\sqrt{15})(7-2\sqrt{10})(5-2\sqrt{6})=\\\\(56-16\sqrt{10}-14\sqrt{15}+4\sqrt{150})(5-2\sqrt{6})=\\\\(56-16\sqrt{10}-14\sqrt{15}+20\sqrt{6})(5-2\sqrt{6})=\\\\280-112\sqrt{6}-80\sqrt{10}+32\sqrt{60}-70\sqrt{15}+28\sqrt{90}+100\sqrt{6}-240=\\\\280-112\sqrt{6}-80\sqrt{10}+64\sqrt{15}-70\sqrt{15}+84\sqrt{10}+100\sqrt{6}-240=\\\\-6\sqrt{15}+4\sqrt{10}-12\sqrt{6}+40$

Calcular la $\frac{x^2x^2yz+y^2y^2xz+z^2z^2xy}{ x^2y^2z^2 }$:

$\frac{ -60\sqrt{15}+28\sqrt{10}-36\sqrt{6}+232+33\sqrt{15}-5\sqrt{10}-51\sqrt{6}+13-11\sqrt{10}+9\sqrt{15}+51\sqrt{6}-125 }{ -6\sqrt{15}+4\sqrt{10}-12\sqrt{6}+40 } = \\\\\frac{ -18\sqrt{15}+12\sqrt{10}-36\sqrt{6}+120 }{ -6\sqrt{15}+4\sqrt{10}-12\sqrt{6}+40 }=\\\\\frac{ 3(-6\sqrt{15}+4\sqrt{10}-12\sqrt{6}+40) }{ -6\sqrt{15}+4\sqrt{10}-12\sqrt{6}+40 }=\\\\3$

Calcular $\frac{x^2+y^2+z^2}{xy+yz+xz}=\frac{-2(\sqrt{15}+\sqrt{10}+\sqrt{6}-10 ) }{ \sqrt{15}+\sqrt{10}+\sqrt{6}-10 }$

$\frac{x^2+y^2+z^2}{xy+yz+xz}= -2$

Calcular E:

$E = (-2)(3) = -6$