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z^4+8-8sqrt(3)i+1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 4           ___          
z  + 8 - 8*\/ 3 *I + 1 = 0
$$\left(\left(z^{4} + 8\right) - 8 \sqrt{3} i\right) + 1 = 0$$
Simplificar
Solución detallada
Tenemos la ecuación
$$\left(\left(z^{4} + 8\right) - 8 \sqrt{3} i\right) + 1 = 0$$
Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -9 + 8*i*sqrt(3) 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:
$$w^{4} = -9 + 8 \sqrt{3} i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{4} e^{4 i p} = -9 + 8 \sqrt{3} i$$
donde
$$r = \sqrt[8]{273}$$
- módulo del número complejo
Sustituyamos r:
$$e^{4 i p} = \frac{\sqrt{273} \left(-9 + 8 \sqrt{3} i\right)}{273}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = \frac{\sqrt{273} \left(-9 + 8 \sqrt{3} i\right)}{273}$$
es decir
$$\cos{\left(4 p \right)} = - \frac{3 \sqrt{273}}{91}$$
y
$$\sin{\left(4 p \right)} = \frac{8 \sqrt{91}}{91}$$
entonces
$$p = \frac{\pi N}{2} - \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4}$$
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:
$$w_{1} = - \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{2} = \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{3} = - \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$w_{4} = \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$

Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{2} = \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{3} = - \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
$$z_{4} = \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$
Gráfica
Respuesta rápida [src] 
                  /      /    ___\     \                /      /    ___\     \
                  |      |8*\/ 3 |     |                |      |8*\/ 3 |     |
                  |  atan|-------|     |                |  atan|-------|     |
       8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|
z1 = - \/ 273 *sin|- ------------- + --| + I*\/ 273 *cos|- ------------- + --|
                  \        4         4 /                \        4         4 /
$$z_{1} = - \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$ 
                /      /    ___\     \                /      /    ___\     \
                |      |8*\/ 3 |     |                |      |8*\/ 3 |     |
                |  atan|-------|     |                |  atan|-------|     |
     8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|
z2 = \/ 273 *sin|- ------------- + --| - I*\/ 273 *cos|- ------------- + --|
                \        4         4 /                \        4         4 /
$$z_{2} = \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$ 
                  /      /    ___\     \                /      /    ___\     \
                  |      |8*\/ 3 |     |                |      |8*\/ 3 |     |
                  |  atan|-------|     |                |  atan|-------|     |
       8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|
z3 = - \/ 273 *cos|- ------------- + --| - I*\/ 273 *sin|- ------------- + --|
                  \        4         4 /                \        4         4 /
$$z_{3} = - \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$ 
                /      /    ___\     \                /      /    ___\     \
                |      |8*\/ 3 |     |                |      |8*\/ 3 |     |
                |  atan|-------|     |                |  atan|-------|     |
     8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|
z4 = \/ 273 *cos|- ------------- + --| + I*\/ 273 *sin|- ------------- + --|
                \        4         4 /                \        4         4 /
$$z_{4} = \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}$$ 
z4 = 273^(1/8)*cos(-atan(8*sqrt(3)/9)/4 + pi/4) + 273^(1/8)*i*sin(-atan(8*sqrt(3)/9)/4 + pi/4)
Suma y producto de raíces [src] 
suma
             /      /    ___\     \                /      /    ___\     \              /      /    ___\     \                /      /    ___\     \                /      /    ___\     \                /      /    ___\     \              /      /    ___\     \                /      /    ___\     \
             |      |8*\/ 3 |     |                |      |8*\/ 3 |     |              |      |8*\/ 3 |     |                |      |8*\/ 3 |     |                |      |8*\/ 3 |     |                |      |8*\/ 3 |     |              |      |8*\/ 3 |     |                |      |8*\/ 3 |     |
             |  atan|-------|     |                |  atan|-------|     |              |  atan|-------|     |                |  atan|-------|     |                |  atan|-------|     |                |  atan|-------|     |              |  atan|-------|     |                |  atan|-------|     |
  8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|   8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|   8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|
- \/ 273 *sin|- ------------- + --| + I*\/ 273 *cos|- ------------- + --| + \/ 273 *sin|- ------------- + --| - I*\/ 273 *cos|- ------------- + --| + - \/ 273 *cos|- ------------- + --| - I*\/ 273 *sin|- ------------- + --| + \/ 273 *cos|- ------------- + --| + I*\/ 273 *sin|- ------------- + --|
             \        4         4 /                \        4         4 /              \        4         4 /                \        4         4 /                \        4         4 /                \        4         4 /              \        4         4 /                \        4         4 /
$$\left(\left(- \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right) + \left(\left(\sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right) + \left(- \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right)\right)\right) + \left(\sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right)$$ 
=
0
$$0$$ 
producto
/             /      /    ___\     \                /      /    ___\     \\ /           /      /    ___\     \                /      /    ___\     \\ /             /      /    ___\     \                /      /    ___\     \\ /           /      /    ___\     \                /      /    ___\     \\
|             |      |8*\/ 3 |     |                |      |8*\/ 3 |     || |           |      |8*\/ 3 |     |                |      |8*\/ 3 |     || |             |      |8*\/ 3 |     |                |      |8*\/ 3 |     || |           |      |8*\/ 3 |     |                |      |8*\/ 3 |     ||
|             |  atan|-------|     |                |  atan|-------|     || |           |  atan|-------|     |                |  atan|-------|     || |             |  atan|-------|     |                |  atan|-------|     || |           |  atan|-------|     |                |  atan|-------|     ||
|  8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|| |8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|| |  8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi|| |8 _____    |      \   9   /   pi|     8 _____    |      \   9   /   pi||
|- \/ 273 *sin|- ------------- + --| + I*\/ 273 *cos|- ------------- + --||*|\/ 273 *sin|- ------------- + --| - I*\/ 273 *cos|- ------------- + --||*|- \/ 273 *cos|- ------------- + --| - I*\/ 273 *sin|- ------------- + --||*|\/ 273 *cos|- ------------- + --| + I*\/ 273 *sin|- ------------- + --||
\             \        4         4 /                \        4         4 // \           \        4         4 /                \        4         4 // \             \        4         4 /                \        4         4 // \           \        4         4 /                \        4         4 //
$$\left(- \sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(\sqrt[8]{273} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(- \sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(\sqrt[8]{273} \cos{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + \sqrt[8]{273} i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right)$$ 
=
                                                               2                                                          2
        /     /      /    ___\     \      /         /    ___\\\  /       /         /    ___\\      /      /    ___\     \\ 
        |     |      |8*\/ 3 |     |      |         |8*\/ 3 |||  |       |         |8*\/ 3 ||      |      |8*\/ 3 |     || 
        |     |  atan|-------|     |      |     atan|-------|||  |       |     atan|-------||      |  atan|-------|     || 
  _____ |     |      \   9   /   pi|      |pi       \   9   /||  |       |pi       \   9   /|      |      \   9   /   pi|| 
\/ 273 *|I*sin|- ------------- + --| + sin|-- + -------------|| *|- I*sin|-- + -------------| + sin|- ------------- + --|| 
        \     \        4         4 /      \4          4      //  \       \4          4      /      \        4         4 //
$$\sqrt{273} \left(\sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} - i \sin{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right)^{2} \left(\sin{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)} + i \sin{\left(- \frac{\operatorname{atan}{\left(\frac{8 \sqrt{3}}{9} \right)}}{4} + \frac{\pi}{4} \right)}\right)^{2}$$ 
sqrt(273)*(i*sin(-atan(8*sqrt(3)/9)/4 + pi/4) + sin(pi/4 + atan(8*sqrt(3)/9)/4))^2*(-i*sin(pi/4 + atan(8*sqrt(3)/9)/4) + sin(-atan(8*sqrt(3)/9)/4 + pi/4))^2
Respuesta numérica [src] 
z1 = -1.7326620224252 - 1.0308715569218*i
z2 = 1.7326620224252 + 1.0308715569218*i
z3 = 1.0308715569218 - 1.7326620224252*i
z4 = -1.0308715569218 + 1.7326620224252*i
z4 = -1.0308715569218 + 1.7326620224252*i
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