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- La ecuación:
-
Ecuación (x-11)^4=(x+3)^4
-
Ecuación √(x^2-1)^3=2*x
-
Ecuación -x/12+x=55/12
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Ecuación 49,86-(0.3+0.049*49,86)=x
- Expresar {x} en función de y en la ecuación:
- 18*x-11*y=2
- 12*x-14*y=20
- -3*x-20*y=-13
- -16*x+14*y=-9
- Integral de d{x}:
- z^3
- Gráfico de la función y =:
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z^3
- Derivada de:
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z^3
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Expresiones idénticas
- z^ tres = uno -sqrt(tres *j)
- z al cubo es igual a 1 menos raíz cuadrada de (3 multiplicar por j)
- z en el grado tres es igual a uno menos raíz cuadrada de (tres multiplicar por j)
- z^3=1-√(3*j)
- z3=1-sqrt(3*j)
- z3=1-sqrt3*j
- z³=1-sqrt(3*j)
- z en el grado 3=1-sqrt(3*j)
- z^3=1-sqrt(3j)
- z3=1-sqrt(3j)
- z3=1-sqrt3j
- z^3=1-sqrt3j
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Expresiones semejantes
- z^3=1+sqrt(3*j)
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(x^2-4)+sqrt(x^2+x-2)=0
- sqrt(x-3)=5-x
- sqrt(x+4)=x-2
- sqrt(-72-17x)=-x
- sqrt(x)-4/(sqrt(2+x))+sqrt(2+x)=0
z^3=1-sqrt(3*j) la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$z^{3} = 1 - \sqrt{3 i}$$
Ya que la potencia en la ecuación es igual a = 3 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 3 de las dos partes de la ecuación:
Obtenemos:
$$\sqrt[3]{z^{3}} = \sqrt[3]{1 - \sqrt{3} \sqrt{i}}$$
o
$$z = \sqrt[3]{1 - \sqrt{3} \sqrt{i}}$$
Abrimos los paréntesis en el miembro derecho de la ecuación
Obtenemos la respuesta: z = (1 - sqrt(3)*sqrt(i))^(1/3)
Las demás 3 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{3} = 1 - \sqrt{3} \sqrt{i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{3} e^{3 i p} = 1 - \sqrt{3} \sqrt{i}$$
donde
$$r = \sqrt[6]{- \frac{\sqrt{6}}{2} + 1 - \frac{3 \sqrt{2} \sqrt{i} i}{2} - \sqrt{3} \sqrt{i} + \frac{\sqrt{6} i}{2} + \frac{3 \sqrt{2} \sqrt{i}}{2}}$$
- módulo del número complejo
Sustituyamos r:
$$e^{3 i p} = \frac{1 - \sqrt{3} \sqrt{i}}{\sqrt{- \frac{\sqrt{6}}{2} + 1 - \frac{3 \sqrt{2} \sqrt{i} i}{2} - \sqrt{3} \sqrt{i} + \frac{\sqrt{6} i}{2} + \frac{3 \sqrt{2} \sqrt{i}}{2}}}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = \frac{1 - \sqrt{3} \sqrt{i}}{\sqrt{- \frac{\sqrt{6}}{2} + 1 - \frac{3 \sqrt{2} \sqrt{i} i}{2} - \sqrt{3} \sqrt{i} + \frac{\sqrt{6} i}{2} + \frac{3 \sqrt{2} \sqrt{i}}{2}}}$$
es decir
$$\cos{\left(3 p \right)} = \frac{1 - \frac{\sqrt{6}}{2}}{\sqrt{4 - \sqrt{6}}}$$
y
$$\sin{\left(3 p \right)} = - \frac{\sqrt{6}}{2 \sqrt{4 - \sqrt{6}}}$$
entonces
$$p = \frac{2 \pi N}{3} - \frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para w
Es decir, la solución será para w:
$$w_{1} = \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} - i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}$$
$$w_{2} = - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
$$w_{3} = - \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} - i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}$$
$$z_{2} = - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
$$z_{3} = - \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
$$z^{3} = 1 - \sqrt{3 i}$$
Ya que la potencia en la ecuación es igual a = 3 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 3 de las dos partes de la ecuación:
Obtenemos:
$$\sqrt[3]{z^{3}} = \sqrt[3]{1 - \sqrt{3} \sqrt{i}}$$
o
$$z = \sqrt[3]{1 - \sqrt{3} \sqrt{i}}$$
Abrimos los paréntesis en el miembro derecho de la ecuación
z = 1+sqrt+3sqrti)^1/3
Obtenemos la respuesta: z = (1 - sqrt(3)*sqrt(i))^(1/3)
Las demás 3 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{3} = 1 - \sqrt{3} \sqrt{i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{3} e^{3 i p} = 1 - \sqrt{3} \sqrt{i}$$
donde
$$r = \sqrt[6]{- \frac{\sqrt{6}}{2} + 1 - \frac{3 \sqrt{2} \sqrt{i} i}{2} - \sqrt{3} \sqrt{i} + \frac{\sqrt{6} i}{2} + \frac{3 \sqrt{2} \sqrt{i}}{2}}$$
- módulo del número complejo
Sustituyamos r:
$$e^{3 i p} = \frac{1 - \sqrt{3} \sqrt{i}}{\sqrt{- \frac{\sqrt{6}}{2} + 1 - \frac{3 \sqrt{2} \sqrt{i} i}{2} - \sqrt{3} \sqrt{i} + \frac{\sqrt{6} i}{2} + \frac{3 \sqrt{2} \sqrt{i}}{2}}}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = \frac{1 - \sqrt{3} \sqrt{i}}{\sqrt{- \frac{\sqrt{6}}{2} + 1 - \frac{3 \sqrt{2} \sqrt{i} i}{2} - \sqrt{3} \sqrt{i} + \frac{\sqrt{6} i}{2} + \frac{3 \sqrt{2} \sqrt{i}}{2}}}$$
es decir
$$\cos{\left(3 p \right)} = \frac{1 - \frac{\sqrt{6}}{2}}{\sqrt{4 - \sqrt{6}}}$$
y
$$\sin{\left(3 p \right)} = - \frac{\sqrt{6}}{2 \sqrt{4 - \sqrt{6}}}$$
entonces
$$p = \frac{2 \pi N}{3} - \frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para w
Es decir, la solución será para w:
$$w_{1} = \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} - i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}$$
$$w_{2} = - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
$$w_{3} = - \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} - i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}$$
$$z_{2} = - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
$$z_{3} = - \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt{3} i \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p z^{2} + q z + v + z^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -1 + \sqrt{3 i}$$
Fórmulas de Cardano-Vieta
$$z_{1} + z_{2} + z_{3} = - p$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = q$$
$$z_{1} z_{2} z_{3} = v$$
$$z_{1} + z_{2} + z_{3} = 0$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = 0$$
$$z_{1} z_{2} z_{3} = -1 + \sqrt{3 i}$$
$$p z^{2} + q z + v + z^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -1 + \sqrt{3 i}$$
Fórmulas de Cardano-Vieta
$$z_{1} + z_{2} + z_{3} = - p$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = q$$
$$z_{1} z_{2} z_{3} = v$$
$$z_{1} + z_{2} + z_{3} = 0$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = 0$$
$$z_{1} z_{2} z_{3} = -1 + \sqrt{3 i}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\ / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | \/ 6 || | | \/ 6 ||| | | \/ 6 || | | \/ 6 || | | | \/ 6 || | | \/ 6 ||| | | \/ 6 || | | \/ 6 ||
/ / ___ \\ / / ___ \\ | | atan|-------------|| | atan|-------------||| | atan|-------------|| | atan|-------------|| | | atan|-------------|| | atan|-------------||| | atan|-------------|| | atan|-------------||
| | \/ 6 || | | \/ 6 || | | | / ___\|| | | / ___\||| | | / ___\|| | | / ___\|| | | | / ___\|| | | / ___\||| | | / ___\|| | | / ___\||
| atan|-------------|| | atan|-------------|| | | | | \/ 6 ||| | | | \/ 6 |||| | | | \/ 6 ||| | | | \/ 6 ||| | | | | \/ 6 ||| | | | \/ 6 |||| | | | \/ 6 ||| | | | \/ 6 |||
| | / ___\|| | | / ___\|| | ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----||| | ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||
| | | \/ 6 ||| | | | \/ 6 ||| |6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| 6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //| |6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| 6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|
___________ | |2*|1 - -----||| ___________ | |2*|1 - -----||| |\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| \/ 4 - \/ 6 *cos|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *sin|-- + -------------------| |\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| \/ 4 - \/ 6 *cos|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *sin|-- + -------------------|
6 / ___ |pi \ \ 2 //| 6 / ___ |pi \ \ 2 //| | \3 3 / \3 3 /| \3 3 / \3 3 / | \3 3 / \3 3 /| \3 3 / \3 3 /
\/ 4 - \/ 6 *cos|-- + -------------------| - I*\/ 4 - \/ 6 *sin|-- + -------------------| + I*|-------------------------------------------- + --------------------------------------------------| - -------------------------------------------- + -------------------------------------------------- + I*|-------------------------------------------- - --------------------------------------------------| - -------------------------------------------- - --------------------------------------------------
\3 3 / \3 3 / \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(- \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + i \left(- \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right)\right) + \left(\left(\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} - i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}\right) + \left(- \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + i \left(\frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right)\right)\right)$$
=
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | | \/ 6 || | | \/ 6 ||| | | | \/ 6 || | | \/ 6 ||| | | atan|-------------|| | atan|-------------||| | | atan|-------------|| | atan|-------------||| / / ___ \\ | | | / ___\|| | | / ___\||| | | | / ___\|| | | / ___\||| | | \/ 6 || | | | | \/ 6 ||| | | | \/ 6 |||| | | | | \/ 6 ||| | | | \/ 6 |||| | atan|-------------|| | ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| | ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| | | / ___\|| |6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| |6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| | | | \/ 6 ||| |\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| |\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| ___________ | |2*|1 - -----||| | \3 3 / \3 3 /| | \3 3 / \3 3 /| 6 / ___ |pi \ \ 2 //| I*|-------------------------------------------- + --------------------------------------------------| + I*|-------------------------------------------- - --------------------------------------------------| - I*\/ 4 - \/ 6 *sin|-- + -------------------| \ 2 2 / \ 2 2 / \3 3 /
$$- i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} + i \left(- \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right) + i \left(\frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right)$$
producto
/ / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\\ / / / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\\
| | | | \/ 6 || | | \/ 6 ||| | | \/ 6 || | | \/ 6 ||| | | | | \/ 6 || | | \/ 6 ||| | | \/ 6 || | | \/ 6 |||
/ / / ___ \\ / / ___ \\\ | | | atan|-------------|| | atan|-------------||| | atan|-------------|| | atan|-------------||| | | | atan|-------------|| | atan|-------------||| | atan|-------------|| | atan|-------------|||
| | | \/ 6 || | | \/ 6 ||| | | | | / ___\|| | | / ___\||| | | / ___\|| | | / ___\||| | | | | / ___\|| | | / ___\||| | | / ___\|| | | / ___\|||
| | atan|-------------|| | atan|-------------||| | | | | | \/ 6 ||| | | | \/ 6 |||| | | | \/ 6 ||| | | | \/ 6 |||| | | | | | \/ 6 ||| | | | \/ 6 |||| | | | \/ 6 ||| | | | \/ 6 ||||
| | | / ___\|| | | / ___\||| | | ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| | | ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----||||
| | | | \/ 6 ||| | | | \/ 6 |||| | |6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| 6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| | |6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| 6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //||
| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| | |\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| \/ 4 - \/ 6 *cos|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *sin|-- + -------------------|| | |\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| \/ 4 - \/ 6 *cos|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *sin|-- + -------------------||
|6 / ___ |pi \ \ 2 //| 6 / ___ |pi \ \ 2 //|| | | \3 3 / \3 3 /| \3 3 / \3 3 /| | | \3 3 / \3 3 /| \3 3 / \3 3 /|
|\/ 4 - \/ 6 *cos|-- + -------------------| - I*\/ 4 - \/ 6 *sin|-- + -------------------||*|I*|-------------------------------------------- + --------------------------------------------------| - -------------------------------------------- + --------------------------------------------------|*|I*|-------------------------------------------- - --------------------------------------------------| - -------------------------------------------- - --------------------------------------------------|
\ \3 3 / \3 3 // \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} - i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}\right) \left(- \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + i \left(\frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right)\right) \left(- \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + i \left(- \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right)\right)$$
=
___________ / / / ___\ \ / / ___\ \ / / ___\ \ / / ___\ \ / / ___\ \ / / ___\ \\
/ ___ | 3| atan\3 + \/ 6 / pi| 3| atan\3 + \/ 6 / pi| 2| atan\3 + \/ 6 / pi| | atan\3 + \/ 6 / pi| 2| atan\3 + \/ 6 / pi| | atan\3 + \/ 6 / pi||
\/ 4 - \/ 6 *|cos |- --------------- + --| + I*sin |- --------------- + --| - 3*sin |- --------------- + --|*cos|- --------------- + --| - 3*I*cos |- --------------- + --|*sin|- --------------- + --||
\ \ 3 3 / \ 3 3 / \ 3 3 / \ 3 3 / \ 3 3 / \ 3 3 //
$$\sqrt{4 - \sqrt{6}} \left(- 3 \sin^{2}{\left(- \frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{3} + \frac{\pi}{3} \right)} \cos{\left(- \frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{3} + \frac{\pi}{3} \right)} + \cos^{3}{\left(- \frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{3} + \frac{\pi}{3} \right)} - 3 i \sin{\left(- \frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{3} + \frac{\pi}{3} \right)} \cos^{2}{\left(- \frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{3} + \frac{\pi}{3} \right)} + i \sin^{3}{\left(- \frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{3} + \frac{\pi}{3} \right)}\right)$$
sqrt(4 - sqrt(6))*(cos(-atan(3 + sqrt(6))/3 + pi/3)^3 + i*sin(-atan(3 + sqrt(6))/3 + pi/3)^3 - 3*sin(-atan(3 + sqrt(6))/3 + pi/3)^2*cos(-atan(3 + sqrt(6))/3 + pi/3) - 3*i*cos(-atan(3 + sqrt(6))/3 + pi/3)^2*sin(-atan(3 + sqrt(6))/3 + pi/3))
Respuesta rápida
[src]
/ / ___ \\ / / ___ \\
| | \/ 6 || | | \/ 6 ||
| atan|-------------|| | atan|-------------||
| | / ___\|| | | / ___\||
| | | \/ 6 ||| | | | \/ 6 |||
___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||
6 / ___ |pi \ \ 2 //| 6 / ___ |pi \ \ 2 //|
z1 = \/ 4 - \/ 6 *cos|-- + -------------------| - I*\/ 4 - \/ 6 *sin|-- + -------------------|
\3 3 / \3 3 /
$$z_{1} = \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)} - i \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}$$
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | \/ 6 || | | \/ 6 ||| | | \/ 6 || | | \/ 6 ||
| | atan|-------------|| | atan|-------------||| | atan|-------------|| | atan|-------------||
| | | / ___\|| | | / ___\||| | | / ___\|| | | / ___\||
| | | | \/ 6 ||| | | | \/ 6 |||| | | | \/ 6 ||| | | | \/ 6 |||
| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||
|6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| 6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|
|\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| \/ 4 - \/ 6 *cos|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *sin|-- + -------------------|
| \3 3 / \3 3 /| \3 3 / \3 3 /
z2 = I*|-------------------------------------------- + --------------------------------------------------| - -------------------------------------------- + --------------------------------------------------
\ 2 2 / 2 2
$$z_{2} = - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + i \left(\frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right)$$
/ / / ___ \\ / / ___ \\\ / / ___ \\ / / ___ \\
| | | \/ 6 || | | \/ 6 ||| | | \/ 6 || | | \/ 6 ||
| | atan|-------------|| | atan|-------------||| | atan|-------------|| | atan|-------------||
| | | / ___\|| | | / ___\||| | | / ___\|| | | / ___\||
| | | | \/ 6 ||| | | | \/ 6 |||| | | | \/ 6 ||| | | | \/ 6 |||
| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||| ___________ | |2*|1 - -----||| ___________ | |2*|1 - -----|||
|6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|| 6 / ___ |pi \ \ 2 //| ___ 6 / ___ |pi \ \ 2 //|
|\/ 4 - \/ 6 *sin|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *cos|-- + -------------------|| \/ 4 - \/ 6 *cos|-- + -------------------| \/ 3 *\/ 4 - \/ 6 *sin|-- + -------------------|
| \3 3 / \3 3 /| \3 3 / \3 3 /
z3 = I*|-------------------------------------------- - --------------------------------------------------| - -------------------------------------------- - --------------------------------------------------
\ 2 2 / 2 2
$$z_{3} = - \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} - \frac{\sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + i \left(- \frac{\sqrt{3} \sqrt[6]{4 - \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2} + \frac{\sqrt[6]{4 - \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(1 - \frac{\sqrt{6}}{2}\right)} \right)}}{3} + \frac{\pi}{3} \right)}}{2}\right)$$
z3 = -sqrt(3)*(4 - sqrt(6))^(1/6)*sin(atan(sqrt(6)/(2*(1 - sqrt(6)/2)))/3 + pi/3)/2 - (4 - sqrt(6))^(1/6)*cos(atan(sqrt(6)/(2*(1 - sqrt(6)/2)))/3 + pi/3)/2 + i*(-sqrt(3)*(4 - sqrt(6))^(1/6)*cos(atan(sqrt(6)/(2*(1 - sqrt(6)/2)))/3 + pi/3)/2 + (4 - sqrt(6))^(1/6)*sin(atan(sqrt(6)/(2*(1 - sqrt(6)/2)))/3 + pi/3)/2)
Respuesta numérica
[src]
z1 = 0.065042722871996 + 1.0738672872497*i
z2 = -0.962517712487319 - 0.48060499328639*i
z3 = 0.897474989615323 - 0.59326229396331*i
z3 = 0.897474989615323 - 0.59326229396331*i