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Expresiones semejantes
- z^4+8*(1+(3*i)^(1/2))=0
- z^4-8*(1-(3*i)^(1/2))=0
z^4+8*(1-(3*i)^(1/2))=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
4 / _____\ z + 8*\1 - \/ 3*I / = 0
$$z^{4} + 8 \left(1 - \sqrt{3 i}\right) = 0$$
Solución detallada
Tenemos la ecuación
$$z^{4} + 8 \left(1 - \sqrt{3 i}\right) = 0$$
Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -8 + 8*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:
$$w^{4} = -8 + 8 \sqrt{3} \sqrt{i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{4} e^{4 i p} = -8 + 8 \sqrt{3} \sqrt{i}$$
donde
$$r = \sqrt[8]{- 32 \sqrt{6} + 64 - 96 \sqrt{2} \sqrt{i} i - 64 \sqrt{3} \sqrt{i} + 32 \sqrt{6} i + 96 \sqrt{2} \sqrt{i}}$$
- módulo del número complejo
Sustituyamos r:
$$e^{4 i p} = \frac{-8 + 8 \sqrt{3} \sqrt{i}}{\sqrt{- 32 \sqrt{6} + 64 - 96 \sqrt{2} \sqrt{i} i - 64 \sqrt{3} \sqrt{i} + 32 \sqrt{6} i + 96 \sqrt{2} \sqrt{i}}}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = \frac{-8 + 8 \sqrt{3} \sqrt{i}}{\sqrt{- 32 \sqrt{6} + 64 - 96 \sqrt{2} \sqrt{i} i - 64 \sqrt{3} \sqrt{i} + 32 \sqrt{6} i + 96 \sqrt{2} \sqrt{i}}}$$
es decir
$$\cos{\left(4 p \right)} = \frac{-8 + 4 \sqrt{6}}{\sqrt{256 - 64 \sqrt{6}}}$$
y
$$\sin{\left(4 p \right)} = \frac{4 \sqrt{6}}{\sqrt{256 - 64 \sqrt{6}}}$$
entonces
$$p = \frac{\pi N}{2} + \frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4}$$
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:
$$w_{1} = - \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$w_{2} = \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$w_{3} = - \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$w_{4} = \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$z_{2} = \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$z_{3} = - \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$z_{4} = \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$z^{4} + 8 \left(1 - \sqrt{3 i}\right) = 0$$
Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -8 + 8*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:
$$w^{4} = -8 + 8 \sqrt{3} \sqrt{i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{4} e^{4 i p} = -8 + 8 \sqrt{3} \sqrt{i}$$
donde
$$r = \sqrt[8]{- 32 \sqrt{6} + 64 - 96 \sqrt{2} \sqrt{i} i - 64 \sqrt{3} \sqrt{i} + 32 \sqrt{6} i + 96 \sqrt{2} \sqrt{i}}$$
- módulo del número complejo
Sustituyamos r:
$$e^{4 i p} = \frac{-8 + 8 \sqrt{3} \sqrt{i}}{\sqrt{- 32 \sqrt{6} + 64 - 96 \sqrt{2} \sqrt{i} i - 64 \sqrt{3} \sqrt{i} + 32 \sqrt{6} i + 96 \sqrt{2} \sqrt{i}}}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = \frac{-8 + 8 \sqrt{3} \sqrt{i}}{\sqrt{- 32 \sqrt{6} + 64 - 96 \sqrt{2} \sqrt{i} i - 64 \sqrt{3} \sqrt{i} + 32 \sqrt{6} i + 96 \sqrt{2} \sqrt{i}}}$$
es decir
$$\cos{\left(4 p \right)} = \frac{-8 + 4 \sqrt{6}}{\sqrt{256 - 64 \sqrt{6}}}$$
y
$$\sin{\left(4 p \right)} = \frac{4 \sqrt{6}}{\sqrt{256 - 64 \sqrt{6}}}$$
entonces
$$p = \frac{\pi N}{2} + \frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4}$$
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:
$$w_{1} = - \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$w_{2} = \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$w_{3} = - \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$w_{4} = \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$z_{2} = \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$z_{3} = - \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
$$z_{4} = \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\
| | 4*\/ 6 || | | 4*\/ 6 || | | 4*\/ 6 || | | 4*\/ 6 || | | 4*\/ 6 || | | 4*\/ 6 || | | 4*\/ 6 || | | 4*\/ 6 ||
|atan|------------|| |atan|------------|| |atan|------------|| |atan|------------|| |atan|------------|| |atan|------------|| |atan|------------|| |atan|------------||
________________ | | ___|| ________________ | | ___|| ________________ | | ___|| ________________ | | ___|| ________________ | | ___|| ________________ | | ___|| ________________ | | ___|| ________________ | | ___||
8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|
- \/ 256 - 64*\/ 6 *sin|------------------| + I*\/ 256 - 64*\/ 6 *cos|------------------| + \/ 256 - 64*\/ 6 *sin|------------------| - I*\/ 256 - 64*\/ 6 *cos|------------------| + - \/ 256 - 64*\/ 6 *cos|------------------| - I*\/ 256 - 64*\/ 6 *sin|------------------| + \/ 256 - 64*\/ 6 *cos|------------------| + I*\/ 256 - 64*\/ 6 *sin|------------------|
\ 4 / \ 4 / \ 4 / \ 4 / \ 4 / \ 4 / \ 4 / \ 4 /
$$\left(\left(- \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right) + \left(\left(\sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right) + \left(- \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right)\right)\right) + \left(\sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right)$$
=
0
$$0$$
producto
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | | 4*\/ 6 || | | 4*\/ 6 ||| | | | 4*\/ 6 || | | 4*\/ 6 ||| | | | 4*\/ 6 || | | 4*\/ 6 ||| | | | 4*\/ 6 || | | 4*\/ 6 ||| | |atan|------------|| |atan|------------||| | |atan|------------|| |atan|------------||| | |atan|------------|| |atan|------------||| | |atan|------------|| |atan|------------||| | ________________ | | ___|| ________________ | | ___||| | ________________ | | ___|| ________________ | | ___||| | ________________ | | ___|| ________________ | | ___||| | ________________ | | ___|| ________________ | | ___||| | 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|| |8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|| | 8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|| |8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|| |- \/ 256 - 64*\/ 6 *sin|------------------| + I*\/ 256 - 64*\/ 6 *cos|------------------||*|\/ 256 - 64*\/ 6 *sin|------------------| - I*\/ 256 - 64*\/ 6 *cos|------------------||*|- \/ 256 - 64*\/ 6 *cos|------------------| - I*\/ 256 - 64*\/ 6 *sin|------------------||*|\/ 256 - 64*\/ 6 *cos|------------------| + I*\/ 256 - 64*\/ 6 *sin|------------------|| \ \ 4 / \ 4 // \ \ 4 / \ 4 // \ \ 4 / \ 4 // \ \ 4 / \ 4 //
$$\left(- \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right) \left(\sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right) \left(- \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right) \left(\sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}\right)$$
=
/ ___\
2 I*atan\3 + \/ 6 /
___________ / / / ___\\ / / ___\\\ -----------------
/ ___ | |atan\3 + \/ 6 /| |atan\3 + \/ 6 /|| 2
8*\/ 4 - \/ 6 *|- sin|---------------| + I*cos|---------------|| *e
\ \ 4 / \ 4 //
$$8 \sqrt{4 - \sqrt{6}} \left(- \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{4} \right)} + i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{4} \right)}\right)^{2} e^{\frac{i \operatorname{atan}{\left(\sqrt{6} + 3 \right)}}{2}}$$
8*sqrt(4 - sqrt(6))*(-sin(atan(3 + sqrt(6))/4) + i*cos(atan(3 + sqrt(6))/4))^2*exp(i*atan(3 + sqrt(6))/2)
Respuesta rápida
[src]
/ / ___ \\ / / ___ \\
| | 4*\/ 6 || | | 4*\/ 6 ||
|atan|------------|| |atan|------------||
________________ | | ___|| ________________ | | ___||
8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|
z1 = - \/ 256 - 64*\/ 6 *sin|------------------| + I*\/ 256 - 64*\/ 6 *cos|------------------|
\ 4 / \ 4 /
$$z_{1} = - \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
/ / ___ \\ / / ___ \\
| | 4*\/ 6 || | | 4*\/ 6 ||
|atan|------------|| |atan|------------||
________________ | | ___|| ________________ | | ___||
8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|
z2 = \/ 256 - 64*\/ 6 *sin|------------------| - I*\/ 256 - 64*\/ 6 *cos|------------------|
\ 4 / \ 4 /
$$z_{2} = \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
/ / ___ \\ / / ___ \\
| | 4*\/ 6 || | | 4*\/ 6 ||
|atan|------------|| |atan|------------||
________________ | | ___|| ________________ | | ___||
8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|
z3 = - \/ 256 - 64*\/ 6 *cos|------------------| - I*\/ 256 - 64*\/ 6 *sin|------------------|
\ 4 / \ 4 /
$$z_{3} = - \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} - i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
/ / ___ \\ / / ___ \\
| | 4*\/ 6 || | | 4*\/ 6 ||
|atan|------------|| |atan|------------||
________________ | | ___|| ________________ | | ___||
8 / ___ | \-8 + 4*\/ 6 /| 8 / ___ | \-8 + 4*\/ 6 /|
z4 = \/ 256 - 64*\/ 6 *cos|------------------| + I*\/ 256 - 64*\/ 6 *sin|------------------|
\ 4 / \ 4 /
$$z_{4} = \sqrt[8]{256 - 64 \sqrt{6}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)} + i \sqrt[8]{256 - 64 \sqrt{6}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{4 \sqrt{6}}{-8 + 4 \sqrt{6}} \right)}}{4} \right)}$$
z4 = (256 - 64*sqrt(6))^(1/8)*cos(atan(4*sqrt(6)/(-8 + 4*sqrt(6)))/4) + i*(256 - 64*sqrt(6))^(1/8)*sin(atan(4*sqrt(6)/(-8 + 4*sqrt(6)))/4)
Respuesta numérica
[src]
z1 = 1.67048124343449 + 0.60471989993659*i
z2 = -1.67048124343449 - 0.60471989993659*i
z3 = -0.60471989993659 + 1.67048124343449*i
z4 = 0.60471989993659 - 1.67048124343449*i
z4 = 0.60471989993659 - 1.67048124343449*i