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- La ecuación:
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Ecuación x=(x+1)/2
- Ecuación (x-6)(-5x-9)=0
-
Ecuación x^2-14*x+33=0
-
Ecuación 5-x^2=0
- Expresar {x} en función de y en la ecuación:
- -17*x-12*y=-9
- 18*x+17*y=-14
- 14*x-10*y=-4
- -17*x-8*y=7
- Derivada de:
-
x^6
- Límite de la función:
-
x^6
- Integral de d{x}:
- x^6
-
Expresiones idénticas
- x^ seis =(uno +i*sqrt(tres))/(- uno +i)
- x en el grado 6 es igual a (1 más i multiplicar por raíz cuadrada de (3)) dividir por ( menos 1 más i)
- x en el grado seis es igual a (uno más i multiplicar por raíz cuadrada de (tres)) dividir por ( menos uno más i)
- x^6=(1+i*√(3))/(-1+i)
- x6=(1+i*sqrt(3))/(-1+i)
- x6=1+i*sqrt3/-1+i
- x⁶=(1+i*sqrt(3))/(-1+i)
- x^6=(1+isqrt(3))/(-1+i)
- x6=(1+isqrt(3))/(-1+i)
- x6=1+isqrt3/-1+i
- x^6=1+isqrt3/-1+i
- x^6=(1+i*sqrt(3)) dividir por (-1+i)
-
Expresiones semejantes
- x^6=(1+i*sqrt(3))/(1+i)
- x^6=(1+i*sqrt(3))/(-1-i)
- x^6=(1-i*sqrt(3))/(-1+i)
-
Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(x-a)+abs(x+a)=4
- sqrt(2*x^2+8*x+7)-2=x
- sqrt^4(17x^2-16)=x
- sqrt(0,5+(sinx)^2)+c0s2x=1
- sqrt4(4x+5-x^2)-sqrt(4x-x^2-3)=1
x^6=(1+i*sqrt(3))/(-1+i) la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
___
6 1 + I*\/ 3
x = -----------
-1 + I
$$x^{6} = \frac{1 + \sqrt{3} i}{-1 + i}$$
Solución detallada
Tenemos la ecuación
$$x^{6} = \frac{1 + \sqrt{3} i}{-1 + i}$$
Ya que la potencia en la ecuación es igual a = 6 y miembro libre = (1 + i*sqrt(3))*(-1 - i)/2 complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 6 raíces son complejas.
hacemos el cambio:
$$z = x$$
entonces la ecuación será así:
$$z^{6} = \frac{\left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{2}$$
Cualquier número complejo se puede presentar que:
$$z = r e^{i p}$$
sustituimos en la ecuación
$$r^{6} e^{6 i p} = \frac{\left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{2}$$
donde
$$r = \sqrt[12]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{6 i p} = \frac{\sqrt{2} \left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{4}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = \frac{\sqrt{2} \left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{4}$$
es decir
$$\cos{\left(6 p \right)} = \frac{\sqrt{2} \left(-1 + \sqrt{3}\right)}{4}$$
y
$$\sin{\left(6 p \right)} = \frac{\sqrt{2} \left(- \sqrt{3} - 1\right)}{4}$$
entonces
$$p = \frac{\pi N}{3} + \frac{\operatorname{atan}{\left(\frac{- \sqrt{3} - 1}{-1 + \sqrt{3}} \right)}}{6}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
$$z_{1} = - \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$z_{2} = \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$z_{3} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$z_{4} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$z_{5} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$z_{6} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
hacemos cambio inverso
$$z = x$$
$$x = z$$
Entonces la respuesta definitiva es:
$$x_{1} = - \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$x_{2} = \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$x_{3} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$x_{4} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$x_{5} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$x_{6} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$x^{6} = \frac{1 + \sqrt{3} i}{-1 + i}$$
Ya que la potencia en la ecuación es igual a = 6 y miembro libre = (1 + i*sqrt(3))*(-1 - i)/2 complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 6 raíces son complejas.
hacemos el cambio:
$$z = x$$
entonces la ecuación será así:
$$z^{6} = \frac{\left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{2}$$
Cualquier número complejo se puede presentar que:
$$z = r e^{i p}$$
sustituimos en la ecuación
$$r^{6} e^{6 i p} = \frac{\left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{2}$$
donde
$$r = \sqrt[12]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{6 i p} = \frac{\sqrt{2} \left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{4}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = \frac{\sqrt{2} \left(-1 - i\right) \left(1 + \sqrt{3} i\right)}{4}$$
es decir
$$\cos{\left(6 p \right)} = \frac{\sqrt{2} \left(-1 + \sqrt{3}\right)}{4}$$
y
$$\sin{\left(6 p \right)} = \frac{\sqrt{2} \left(- \sqrt{3} - 1\right)}{4}$$
entonces
$$p = \frac{\pi N}{3} + \frac{\operatorname{atan}{\left(\frac{- \sqrt{3} - 1}{-1 + \sqrt{3}} \right)}}{6}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
$$z_{1} = - \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$z_{2} = \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$z_{3} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$z_{4} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$z_{5} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$z_{6} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
hacemos cambio inverso
$$z = x$$
$$x = z$$
Entonces la respuesta definitiva es:
$$x_{1} = - \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$x_{2} = \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
$$x_{3} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$x_{4} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$x_{5} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
$$x_{6} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\ / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\ / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\ / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 || | | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 || | | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 || | | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||
/ / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------|| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------|| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------|| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||
| | 1 \/ 3 || | | 1 \/ 3 || | | 1 \/ 3 || | | 1 \/ 3 || | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\|| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\|| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\|| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||
|atan|--------------- + ---------------|| |atan|--------------- + ---------------|| |atan|--------------- + ---------------|| |atan|--------------- + ---------------|| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||
| | / ___\ / ___\|| | | / ___\ / ___\|| | | / ___\ / ___\|| | | / ___\ / ___\|| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||
| | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 ||| |12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //| |12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //| | 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //| | 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|
| |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----||| |\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------| |\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------| | \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------| | \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|
12___ | \ \ 2 2 / \ 2 2 //| 12___ | \ \ 2 2 / \ 2 2 //| 12___ | \ \ 2 2 / \ 2 2 //| 12___ | \ \ 2 2 / \ 2 2 //| | \ 6 / \ 6 /| \ 6 / \ 6 / | \ 6 / \ 6 /| \ 6 / \ 6 / | \ 6 / \ 6 /| \ 6 / \ 6 / | \ 6 / \ 6 /| \ 6 / \ 6 /
- \/ 2 *cos|---------------------------------------| + I*\/ 2 *sin|---------------------------------------| + \/ 2 *cos|---------------------------------------| - I*\/ 2 *sin|---------------------------------------| + I*|-------------------------------------------------- - --------------------------------------------------------| - -------------------------------------------------- - -------------------------------------------------------- + I*|-------------------------------------------------- + --------------------------------------------------------| - -------------------------------------------------- + -------------------------------------------------------- + I*|- -------------------------------------------------- - --------------------------------------------------------| + -------------------------------------------------- - -------------------------------------------------------- + I*|- -------------------------------------------------- + --------------------------------------------------------| + -------------------------------------------------- + --------------------------------------------------------
\ 6 / \ 6 / \ 6 / \ 6 / \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(\left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right) + \left(\left(\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}\right) + \left(- \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}\right)\right)\right) + \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right)\right)\right) + \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right)$$
=
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | | 1 \/ 3 || | | 1 \/ 3 ||| | | | 1 \/ 3 || | | 1 \/ 3 ||| | | | 1 \/ 3 || | | 1 \/ 3 ||| | | | 1 \/ 3 || | | 1 \/ 3 ||| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| |12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| |12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| | 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| | 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| |\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| |\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| | \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| | \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| | \ 6 / \ 6 /| | \ 6 / \ 6 /| | \ 6 / \ 6 /| | \ 6 / \ 6 /| I*|-------------------------------------------------- + --------------------------------------------------------| + I*|-------------------------------------------------- - --------------------------------------------------------| + I*|- -------------------------------------------------- + --------------------------------------------------------| + I*|- -------------------------------------------------- - --------------------------------------------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right) + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)$$
producto
/ / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\\ / / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\\ / / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\\ / / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\\
| | | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||| | | | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||| | | | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||| | | | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 |||
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------|||
| | | 1 \/ 3 || | | 1 \/ 3 ||| | | | 1 \/ 3 || | | 1 \/ 3 ||| | | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||| | | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||| | | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||| | | | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\|||
| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| | | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 ||||
| | | / ___\ / ___\|| | | / ___\ / ___\||| | | | / ___\ / ___\|| | | / ___\ / ___\||| | | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----||||
| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | |12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| | |12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| | | 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| | | 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //||
| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|| | |\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|| | | \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|| | | \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------||
| 12___ | \ \ 2 2 / \ 2 2 //| 12___ | \ \ 2 2 / \ 2 2 //|| |12___ | \ \ 2 2 / \ 2 2 //| 12___ | \ \ 2 2 / \ 2 2 //|| | | \ 6 / \ 6 /| \ 6 / \ 6 /| | | \ 6 / \ 6 /| \ 6 / \ 6 /| | | \ 6 / \ 6 /| \ 6 / \ 6 /| | | \ 6 / \ 6 /| \ 6 / \ 6 /|
|- \/ 2 *cos|---------------------------------------| + I*\/ 2 *sin|---------------------------------------||*|\/ 2 *cos|---------------------------------------| - I*\/ 2 *sin|---------------------------------------||*|I*|-------------------------------------------------- - --------------------------------------------------------| - -------------------------------------------------- - --------------------------------------------------------|*|I*|-------------------------------------------------- + --------------------------------------------------------| - -------------------------------------------------- + --------------------------------------------------------|*|I*|- -------------------------------------------------- - --------------------------------------------------------| + -------------------------------------------------- - --------------------------------------------------------|*|I*|- -------------------------------------------------- + --------------------------------------------------------| + -------------------------------------------------- + --------------------------------------------------------|
\ \ 6 / \ 6 // \ \ 6 / \ 6 // \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(- \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}\right) \left(\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}\right) \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right) \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)\right)$$
=
___ / 6/5*pi\ 4/5*pi\ 2/5*pi\ 6/5*pi\ 3/5*pi\ 3/5*pi\ 5/5*pi\ /5*pi\ 5/5*pi\ /5*pi\\
\/ 2 *|- cos |----| - 45*sin |----| + 15*sin |----| + 31*sin |----| - 20*I*cos |----|*sin |----| + 6*I*cos |----|*sin|----| + 6*I*sin |----|*cos|----||
\ \ 72 / \ 72 / \ 72 / \ 72 / \ 72 / \ 72 / \ 72 / \ 72 / \ 72 / \ 72 //
$$\sqrt{2} \left(- \cos^{6}{\left(\frac{5 \pi}{72} \right)} - 45 \sin^{4}{\left(\frac{5 \pi}{72} \right)} + 31 \sin^{6}{\left(\frac{5 \pi}{72} \right)} + 15 \sin^{2}{\left(\frac{5 \pi}{72} \right)} - 20 i \sin^{3}{\left(\frac{5 \pi}{72} \right)} \cos^{3}{\left(\frac{5 \pi}{72} \right)} + 6 i \sin^{5}{\left(\frac{5 \pi}{72} \right)} \cos{\left(\frac{5 \pi}{72} \right)} + 6 i \sin{\left(\frac{5 \pi}{72} \right)} \cos^{5}{\left(\frac{5 \pi}{72} \right)}\right)$$
sqrt(2)*(-cos(5*pi/72)^6 - 45*sin(5*pi/72)^4 + 15*sin(5*pi/72)^2 + 31*sin(5*pi/72)^6 - 20*i*cos(5*pi/72)^3*sin(5*pi/72)^3 + 6*i*cos(5*pi/72)^5*sin(5*pi/72) + 6*i*sin(5*pi/72)^5*cos(5*pi/72))
Respuesta rápida
[src]
/ / ___ \\ / / ___ \\
| | 1 \/ 3 || | | 1 \/ 3 ||
|atan|--------------- + ---------------|| |atan|--------------- + ---------------||
| | / ___\ / ___\|| | | / ___\ / ___\||
| | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||
| |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||
12___ | \ \ 2 2 / \ 2 2 //| 12___ | \ \ 2 2 / \ 2 2 //|
x1 = - \/ 2 *cos|---------------------------------------| + I*\/ 2 *sin|---------------------------------------|
\ 6 / \ 6 /
$$x_{1} = - \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
/ / ___ \\ / / ___ \\
| | 1 \/ 3 || | | 1 \/ 3 ||
|atan|--------------- + ---------------|| |atan|--------------- + ---------------||
| | / ___\ / ___\|| | | / ___\ / ___\||
| | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||
| |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||
12___ | \ \ 2 2 / \ 2 2 //| 12___ | \ \ 2 2 / \ 2 2 //|
x2 = \/ 2 *cos|---------------------------------------| - I*\/ 2 *sin|---------------------------------------|
\ 6 / \ 6 /
$$x_{2} = \sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}$$
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||
| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||
| | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||
| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||
| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||
|12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|
|\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|
| \ 6 / \ 6 /| \ 6 / \ 6 /
x3 = I*|-------------------------------------------------- - --------------------------------------------------------| - -------------------------------------------------- - --------------------------------------------------------
\ 2 2 / 2 2
$$x_{3} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)$$
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||
| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||
| | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||
| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||
| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||
|12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|
|\/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|
| \ 6 / \ 6 /| \ 6 / \ 6 /
x4 = I*|-------------------------------------------------- + --------------------------------------------------------| - -------------------------------------------------- + --------------------------------------------------------
\ 2 2 / 2 2
$$x_{4} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)$$
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||
| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||
| | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||
| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||
| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||
| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|
| \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|
| \ 6 / \ 6 /| \ 6 / \ 6 /
x5 = I*|- -------------------------------------------------- - --------------------------------------------------------| + -------------------------------------------------- - --------------------------------------------------------
\ 2 2 / 2 2
$$x_{5} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)$$
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | 1 \/ 3 || | | 1 \/ 3 ||| | | 1 \/ 3 || | | 1 \/ 3 ||
| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||| |atan|--------------- + ---------------|| |atan|--------------- + ---------------||
| | | / ___\ / ___\|| | | / ___\ / ___\||| | | / ___\ / ___\|| | | / ___\ / ___\||
| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||| | | | 1 \/ 3 | | 1 \/ 3 ||| | | | 1 \/ 3 | | 1 \/ 3 |||
| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||| | |2*|- - + -----| 2*|- - + -----||| | |2*|- - + -----| 2*|- - + -----|||
| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|| 12___ | \ \ 2 2 / \ 2 2 //| 12___ ___ | \ \ 2 2 / \ 2 2 //|
| \/ 2 *sin|---------------------------------------| \/ 2 *\/ 3 *cos|---------------------------------------|| \/ 2 *cos|---------------------------------------| \/ 2 *\/ 3 *sin|---------------------------------------|
| \ 6 / \ 6 /| \ 6 / \ 6 /
x6 = I*|- -------------------------------------------------- + --------------------------------------------------------| + -------------------------------------------------- + --------------------------------------------------------
\ 2 2 / 2 2
$$x_{6} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} + \frac{\sqrt{3}}{2 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)} \right)}}{6} \right)}}{2}\right)$$
x6 = 2^(1/12)*sqrt(3)*sin(atan(1/(2*(-1/2 + sqrt(3)/2)) + sqrt(3)/(2*(-1/2 + sqrt(3)/2)))/6)/2 + 2^(1/12)*cos(atan(1/(2*(-1/2 + sqrt(3)/2)) + sqrt(3)/(2*(-1/2 + sqrt(3)/2)))/6)/2 + i*(-2^(1/12)*sin(atan(1/(2*(-1/2 + sqrt(3)/2)) + sqrt(3)/(2*(-1/2 + sqrt(3)/2)))/6)/2 + 2^(1/12)*sqrt(3)*cos(atan(1/(2*(-1/2 + sqrt(3)/2)) + sqrt(3)/(2*(-1/2 + sqrt(3)/2)))/6)/2)
Respuesta numérica
[src]
x1 = -1.03434958871391 + 0.229309783124794*i
x2 = -0.318586696834583 + 1.01042791178263*i
x3 = 0.318586696834583 - 1.01042791178263*i
x4 = -0.715762891879326 - 0.781118128657834*i
x5 = 1.03434958871391 - 0.229309783124794*i
x6 = 0.715762891879326 + 0.781118128657834*i
x6 = 0.715762891879326 + 0.781118128657834*i