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- La ecuación:
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Ecuación x^4=(x-20)^2
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Ecuación 5-2*(x-1)=4-x
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Ecuación (x+6)^3=25*(x+6)
- Ecuación x+y+2=0
- Expresar {x} en función de y en la ecuación:
- 2*x+13*y=19
- -12*x+3*y=-9
- 10*x-18*y=3
- 16*x+7*y=8
- Derivada de:
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z^4
- Gráfico de la función y =:
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z^4
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Expresiones idénticas
- z^ cuatro =-3sqrt(3i)- nueve = cero
- z en el grado 4 es igual a menos 3 raíz cuadrada de (3i) menos 9 es igual a 0
- z en el grado cuatro es igual a menos 3 raíz cuadrada de (3i) menos nueve es igual a cero
- z^4=-3√(3i)-9=0
- z4=-3sqrt(3i)-9=0
- z4=-3sqrt3i-9=0
- z⁴=-3sqrt(3i)-9=0
- z^4=-3sqrt3i-9=0
- z^4=-3sqrt(3i)-9=O
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Expresiones semejantes
- z^4=+3sqrt(3i)-9=0
- z^4=-3sqrt(3i)+9=0
z^4=-3sqrt(3i)-9=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$z^{4} = -9 - 3 \sqrt{3 i}$$
Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -9 - 3*sqrt(3)*sqrt(i) complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 4 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{4} = -9 - 3 \sqrt{3} \sqrt{i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{4} e^{4 i p} = -9 - 3 \sqrt{3} \sqrt{i}$$
donde
$$r = \sqrt[8]{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}$$
- módulo del número complejo
Sustituyamos r:
$$e^{4 i p} = \frac{-9 - 3 \sqrt{3} \sqrt{i}}{\sqrt{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = \frac{-9 - 3 \sqrt{3} \sqrt{i}}{\sqrt{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}}$$
es decir
$$\cos{\left(4 p \right)} = \frac{-9 - \frac{3 \sqrt{6}}{2}}{\sqrt{27 \sqrt{6} + 108}}$$
y
$$\sin{\left(4 p \right)} = - \frac{3 \sqrt{6}}{2 \sqrt{27 \sqrt{6} + 108}}$$
entonces
$$p = \frac{\pi N}{2} - \frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4}$$
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:
$$w_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z^{4} = -9 - 3 \sqrt{3 i}$$
Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -9 - 3*sqrt(3)*sqrt(i) complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 4 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{4} = -9 - 3 \sqrt{3} \sqrt{i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{4} e^{4 i p} = -9 - 3 \sqrt{3} \sqrt{i}$$
donde
$$r = \sqrt[8]{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}$$
- módulo del número complejo
Sustituyamos r:
$$e^{4 i p} = \frac{-9 - 3 \sqrt{3} \sqrt{i}}{\sqrt{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = \frac{-9 - 3 \sqrt{3} \sqrt{i}}{\sqrt{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}}$$
es decir
$$\cos{\left(4 p \right)} = \frac{-9 - \frac{3 \sqrt{6}}{2}}{\sqrt{27 \sqrt{6} + 108}}$$
y
$$\sin{\left(4 p \right)} = - \frac{3 \sqrt{6}}{2 \sqrt{27 \sqrt{6} + 108}}$$
entonces
$$p = \frac{\pi N}{2} - \frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4}$$
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:
$$w_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\
| | 3*\/ 6 || | | 3*\/ 6 || | | 3*\/ 6 || | | 3*\/ 6 || | | 3*\/ 6 || | | 3*\/ 6 || | | 3*\/ 6 || | | 3*\/ 6 ||
| atan|----------------|| | atan|----------------|| | atan|----------------|| | atan|----------------|| | atan|----------------|| | atan|----------------|| | atan|----------------|| | atan|----------------||
| | / ___\|| | | / ___\|| | | / ___\|| | | / ___\|| | | / ___\|| | | / ___\|| | | / ___\|| | | / ___\||
| | | 3*\/ 6 ||| | | | 3*\/ 6 ||| | | | 3*\/ 6 ||| | | | 3*\/ 6 ||| | | | 3*\/ 6 ||| | | | 3*\/ 6 ||| | | | 3*\/ 6 ||| | | | 3*\/ 6 |||
________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||
8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|
- \/ 108 + 27*\/ 6 *sin|-- + ----------------------| - I*\/ 108 + 27*\/ 6 *cos|-- + ----------------------| + \/ 108 + 27*\/ 6 *sin|-- + ----------------------| + I*\/ 108 + 27*\/ 6 *cos|-- + ----------------------| + - \/ 108 + 27*\/ 6 *cos|-- + ----------------------| + I*\/ 108 + 27*\/ 6 *sin|-- + ----------------------| + \/ 108 + 27*\/ 6 *cos|-- + ----------------------| - I*\/ 108 + 27*\/ 6 *sin|-- + ----------------------|
\4 4 / \4 4 / \4 4 / \4 4 / \4 4 / \4 4 / \4 4 / \4 4 /
$$\left(\sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) + \left(\left(\left(- \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) + \left(\sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right)\right) + \left(- \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right)\right)$$
=
0
$$0$$
producto
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | | 3*\/ 6 || | | 3*\/ 6 ||| | | | 3*\/ 6 || | | 3*\/ 6 ||| | | | 3*\/ 6 || | | 3*\/ 6 ||| | | | 3*\/ 6 || | | 3*\/ 6 ||| | | atan|----------------|| | atan|----------------||| | | atan|----------------|| | atan|----------------||| | | atan|----------------|| | atan|----------------||| | | atan|----------------|| | atan|----------------||| | | | / ___\|| | | / ___\||| | | | / ___\|| | | / ___\||| | | | / ___\|| | | / ___\||| | | | / ___\|| | | / ___\||| | | | | 3*\/ 6 ||| | | | 3*\/ 6 |||| | | | | 3*\/ 6 ||| | | | 3*\/ 6 |||| | | | | 3*\/ 6 ||| | | | 3*\/ 6 |||| | | | | 3*\/ 6 ||| | | | 3*\/ 6 |||| | ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||| | ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||| | ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||| | ________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||| | 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|| |8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|| | 8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|| |8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|| |- \/ 108 + 27*\/ 6 *sin|-- + ----------------------| - I*\/ 108 + 27*\/ 6 *cos|-- + ----------------------||*|\/ 108 + 27*\/ 6 *sin|-- + ----------------------| + I*\/ 108 + 27*\/ 6 *cos|-- + ----------------------||*|- \/ 108 + 27*\/ 6 *cos|-- + ----------------------| + I*\/ 108 + 27*\/ 6 *sin|-- + ----------------------||*|\/ 108 + 27*\/ 6 *cos|-- + ----------------------| - I*\/ 108 + 27*\/ 6 *sin|-- + ----------------------|| \ \4 4 / \4 4 // \ \4 4 / \4 4 // \ \4 4 / \4 4 // \ \4 4 / \4 4 //
$$\left(- \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(\sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(- \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(\sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right)$$
=
/ / ___\\
2 | |1 \/ 6 ||
/ / / ___\\ / / ___\\\ | atan|- - -----||
| | |1 \/ 6 || | |1 \/ 6 ||| | pi \5 5 /|
______________ | | atan|- - -----|| | atan|- - -----||| 2*I*|- -- - ---------------|
/ ___ | |pi \5 5 /| |pi \5 5 /|| \ 4 4 /
3*\/ 12 + 3*\/ 6 *|I*cos|-- + ---------------| + sin|-- + ---------------|| *e
\ \4 4 / \4 4 //
$$3 \sqrt{3 \sqrt{6} + 12} \left(\sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{6}}{5} \right)}}{4} + \frac{\pi}{4} \right)} + i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{6}}{5} \right)}}{4} + \frac{\pi}{4} \right)}\right)^{2} e^{2 i \left(- \frac{\pi}{4} - \frac{\operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{6}}{5} \right)}}{4}\right)}$$
3*sqrt(12 + 3*sqrt(6))*(i*cos(pi/4 + atan(1/5 - sqrt(6)/5)/4) + sin(pi/4 + atan(1/5 - sqrt(6)/5)/4))^2*exp(2*i*(-pi/4 - atan(1/5 - sqrt(6)/5)/4))
Respuesta rápida
[src]
/ / ___ \\ / / ___ \\
| | 3*\/ 6 || | | 3*\/ 6 ||
| atan|----------------|| | atan|----------------||
| | / ___\|| | | / ___\||
| | | 3*\/ 6 ||| | | | 3*\/ 6 |||
________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||
8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|
z1 = - \/ 108 + 27*\/ 6 *sin|-- + ----------------------| - I*\/ 108 + 27*\/ 6 *cos|-- + ----------------------|
\4 4 / \4 4 /
$$z_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
/ / ___ \\ / / ___ \\
| | 3*\/ 6 || | | 3*\/ 6 ||
| atan|----------------|| | atan|----------------||
| | / ___\|| | | / ___\||
| | | 3*\/ 6 ||| | | | 3*\/ 6 |||
________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||
8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|
z2 = \/ 108 + 27*\/ 6 *sin|-- + ----------------------| + I*\/ 108 + 27*\/ 6 *cos|-- + ----------------------|
\4 4 / \4 4 /
$$z_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
/ / ___ \\ / / ___ \\
| | 3*\/ 6 || | | 3*\/ 6 ||
| atan|----------------|| | atan|----------------||
| | / ___\|| | | / ___\||
| | | 3*\/ 6 ||| | | | 3*\/ 6 |||
________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||
8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|
z3 = - \/ 108 + 27*\/ 6 *cos|-- + ----------------------| + I*\/ 108 + 27*\/ 6 *sin|-- + ----------------------|
\4 4 / \4 4 /
$$z_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
/ / ___ \\ / / ___ \\
| | 3*\/ 6 || | | 3*\/ 6 ||
| atan|----------------|| | atan|----------------||
| | / ___\|| | | / ___\||
| | | 3*\/ 6 ||| | | | 3*\/ 6 |||
________________ | |2*|-9 - -------||| ________________ | |2*|-9 - -------|||
8 / ___ |pi \ \ 2 //| 8 / ___ |pi \ \ 2 //|
z4 = \/ 108 + 27*\/ 6 *cos|-- + ----------------------| - I*\/ 108 + 27*\/ 6 *sin|-- + ----------------------|
\4 4 / \4 4 /
$$z_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}$$
z4 = (27*sqrt(6) + 108)^(1/8)*cos(atan(3*sqrt(6)/(2*(-9 - 3*sqrt(6)/2)))/4 + pi/4) - i*(27*sqrt(6) + 108)^(1/8)*sin(atan(3*sqrt(6)/(2*(-9 - 3*sqrt(6)/2)))/4 + pi/4)
Respuesta numérica
[src]
z1 = -1.24936764883673 - 1.43934700597303*i
z2 = 1.24936764883673 + 1.43934700597303*i
z3 = 1.43934700597303 - 1.24936764883673*i
z4 = -1.43934700597303 + 1.24936764883673*i
z4 = -1.43934700597303 + 1.24936764883673*i