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z^6-sqrt3-i=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 6     ___        
z  - \/ 3  - I = 0
(z6−3)−i=0\left(z^{6} - \sqrt{3}\right) - i = 0
Simplificar
Solución detallada
Tenemos la ecuación
(z6−3)−i=0\left(z^{6} - \sqrt{3}\right) - i = 0
Ya que la potencia en la ecuación es igual a = 6 y miembro libre = i + sqrt(3) complejo,
significa que la ecuación correspondiente no tiene soluciones reales

Las demás 6 raíces son complejas.
hacemos el cambio:
w=zw = z
entonces la ecuación será así:
w6=3+iw^{6} = \sqrt{3} + i
Cualquier número complejo se puede presentar que:
w=reipw = r e^{i p}
sustituimos en la ecuación
r6e6ip=3+ir^{6} e^{6 i p} = \sqrt{3} + i
donde
r=26r = \sqrt[6]{2}
- módulo del número complejo
Sustituyamos r:
e6ip=32+i2e^{6 i p} = \frac{\sqrt{3}}{2} + \frac{i}{2}
Usando la fórmula de Euler hallemos las raíces para p
isin⁡(6p)+cos⁡(6p)=32+i2i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = \frac{\sqrt{3}}{2} + \frac{i}{2}
es decir
cos⁡(6p)=32\cos{\left(6 p \right)} = \frac{\sqrt{3}}{2}
y
sin⁡(6p)=12\sin{\left(6 p \right)} = \frac{1}{2}
entonces
p=πN3+π36p = \frac{\pi N}{3} + \frac{\pi}{36}
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:
w1=−26cos⁡(π36)−26isin⁡(π36)w_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}
w2=26cos⁡(π36)+26isin⁡(π36)w_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}
w3=−26cos⁡(π36)2−263sin⁡(π36)2−26isin⁡(π36)2+263icos⁡(π36)2w_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}
w4=−26cos⁡(π36)2+263sin⁡(π36)2−263icos⁡(π36)2−26isin⁡(π36)2w_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}
w5=−263sin⁡(π36)2+26cos⁡(π36)2+26isin⁡(π36)2+263icos⁡(π36)2w_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}
w6=263sin⁡(π36)2+26cos⁡(π36)2−263icos⁡(π36)2+26isin⁡(π36)2w_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}
hacemos cambio inverso
w=zw = z
z=wz = w

Entonces la respuesta definitiva es:
z1=−26cos⁡(π36)−26isin⁡(π36)z_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}
z2=26cos⁡(π36)+26isin⁡(π36)z_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}
z3=−26cos⁡(π36)2−263sin⁡(π36)2−26isin⁡(π36)2+263icos⁡(π36)2z_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}
z4=−26cos⁡(π36)2+263sin⁡(π36)2−263icos⁡(π36)2−26isin⁡(π36)2z_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}
z5=−263sin⁡(π36)2+26cos⁡(π36)2+26isin⁡(π36)2+263icos⁡(π36)2z_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}
z6=263sin⁡(π36)2+26cos⁡(π36)2−263icos⁡(π36)2+26isin⁡(π36)2z_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}
Gráfica
Suma y producto de raíces [src] 
suma
                                                                        /  6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\     /  6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\     /6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\     /6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\
                                                                        |  \/ 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|--|
  6 ___    /pi\     6 ___    /pi\   6 ___    /pi\     6 ___    /pi\     |           \36/                  \36/|            \36/                  \36/     |           \36/                  \36/|            \36/                  \36/     |         \36/                  \36/|            \36/                  \36/     |         \36/                  \36/|            \36/                  \36/
- \/ 2 *cos|--| - I*\/ 2 *sin|--| + \/ 2 *cos|--| + I*\/ 2 *sin|--| + I*|- ------------- + -------------------| - ------------- - ------------------- + I*|- ------------- - -------------------| - ------------- + ------------------- + I*|------------- + -------------------| + ------------- - ------------------- + I*|------------- - -------------------| + ------------- + -------------------
           \36/              \36/            \36/              \36/     \        2                  2         /         2                  2              \        2                  2         /         2                  2              \      2                  2         /         2                  2              \      2                  2         /         2                  2
(263sin⁡(π36)2+26cos⁡(π36)2+i(−263cos⁡(π36)2+26sin⁡(π36)2))+(((−26cos⁡(π36)2+263sin⁡(π36)2+i(−263cos⁡(π36)2−26sin⁡(π36)2))+(((−26cos⁡(π36)−26isin⁡(π36))+(26cos⁡(π36)+26isin⁡(π36)))+(−26cos⁡(π36)2−263sin⁡(π36)2+i(−26sin⁡(π36)2+263cos⁡(π36)2))))+(−263sin⁡(π36)2+26cos⁡(π36)2+i(26sin⁡(π36)2+263cos⁡(π36)2)))\left(\frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right) + \left(\left(\left(- \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right) + \left(\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right)\right) + \left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right)\right)\right) + \left(- \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right)\right) 
=
  /6 ___    /pi\   6 ___   ___    /pi\\     /6 ___    /pi\   6 ___   ___    /pi\\     /  6 ___    /pi\   6 ___   ___    /pi\\     /  6 ___    /pi\   6 ___   ___    /pi\\
  |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||     |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||     |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||     |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||
  |         \36/                  \36/|     |         \36/                  \36/|     |           \36/                  \36/|     |           \36/                  \36/|
I*|------------- + -------------------| + I*|------------- - -------------------| + I*|- ------------- + -------------------| + I*|- ------------- - -------------------|
  \      2                  2         /     \      2                  2         /     \        2                  2         /     \        2                  2         /
i(−263cos⁡(π36)2−26sin⁡(π36)2)+i(−263cos⁡(π36)2+26sin⁡(π36)2)+i(−26sin⁡(π36)2+263cos⁡(π36)2)+i(26sin⁡(π36)2+263cos⁡(π36)2)i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right) + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right) 
producto
                                                                      /  /  6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\\ /  /  6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\\ /  /6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\\ /  /6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\\
                                                                      |  |  \/ 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|--||
/  6 ___    /pi\     6 ___    /pi\\ /6 ___    /pi\     6 ___    /pi\\ |  |           \36/                  \36/|            \36/                  \36/| |  |           \36/                  \36/|            \36/                  \36/| |  |         \36/                  \36/|            \36/                  \36/| |  |         \36/                  \36/|            \36/                  \36/|
|- \/ 2 *cos|--| - I*\/ 2 *sin|--||*|\/ 2 *cos|--| + I*\/ 2 *sin|--||*|I*|- ------------- + -------------------| - ------------- - -------------------|*|I*|- ------------- - -------------------| - ------------- + -------------------|*|I*|------------- + -------------------| + ------------- - -------------------|*|I*|------------- - -------------------| + ------------- + -------------------|
\           \36/              \36// \         \36/              \36// \  \        2                  2         /         2                  2         / \  \        2                  2         /         2                  2         / \  \      2                  2         /         2                  2         / \  \      2                  2         /         2                  2         /
(−26cos⁡(π36)−26isin⁡(π36))(26cos⁡(π36)+26isin⁡(π36))(−26cos⁡(π36)2−263sin⁡(π36)2+i(−26sin⁡(π36)2+263cos⁡(π36)2))(−26cos⁡(π36)2+263sin⁡(π36)2+i(−263cos⁡(π36)2−26sin⁡(π36)2))(−263sin⁡(π36)2+26cos⁡(π36)2+i(26sin⁡(π36)2+263cos⁡(π36)2))(263sin⁡(π36)2+26cos⁡(π36)2+i(−263cos⁡(π36)2+26sin⁡(π36)2))\left(- \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right) \left(\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right) \left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right) \left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right) \left(- \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right) \left(\frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right) 
=
                                                            pi*I
                            2                            2  ----
   /     /5*pi\      /5*pi\\  /       /7*pi\      /7*pi\\    18 
-2*|I*cos|----| + sin|----|| *|- I*cos|----| + sin|----|| *e    
   \     \ 36 /      \ 36 //  \       \ 36 /      \ 36 //
−2(sin⁡(5π36)+icos⁡(5π36))2(sin⁡(7π36)−icos⁡(7π36))2eiπ18- 2 \left(\sin{\left(\frac{5 \pi}{36} \right)} + i \cos{\left(\frac{5 \pi}{36} \right)}\right)^{2} \left(\sin{\left(\frac{7 \pi}{36} \right)} - i \cos{\left(\frac{7 \pi}{36} \right)}\right)^{2} e^{\frac{i \pi}{18}} 
-2*(i*cos(5*pi/36) + sin(5*pi/36))^2*(-i*cos(7*pi/36) + sin(7*pi/36))^2*exp(pi*i/18)
Respuesta rápida [src] 
       6 ___    /pi\     6 ___    /pi\
z1 = - \/ 2 *cos|--| - I*\/ 2 *sin|--|
                \36/              \36/
z1=−26cos⁡(π36)−26isin⁡(π36)z_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)} 
     6 ___    /pi\     6 ___    /pi\
z2 = \/ 2 *cos|--| + I*\/ 2 *sin|--|
              \36/              \36/
z2=26cos⁡(π36)+26isin⁡(π36)z_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)} 
       /  6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\
       |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
       |           \36/                  \36/|            \36/                  \36/
z3 = I*|- ------------- + -------------------| - ------------- - -------------------
       \        2                  2         /         2                  2
z3=−26cos⁡(π36)2−263sin⁡(π36)2+i(−26sin⁡(π36)2+263cos⁡(π36)2)z_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right) 
       /  6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\
       |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
       |           \36/                  \36/|            \36/                  \36/
z4 = I*|- ------------- - -------------------| - ------------- + -------------------
       \        2                  2         /         2                  2
z4=−26cos⁡(π36)2+263sin⁡(π36)2+i(−263cos⁡(π36)2−26sin⁡(π36)2)z_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right) 
       /6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\
       |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
       |         \36/                  \36/|            \36/                  \36/
z5 = I*|------------- + -------------------| + ------------- - -------------------
       \      2                  2         /         2                  2
z5=−263sin⁡(π36)2+26cos⁡(π36)2+i(26sin⁡(π36)2+263cos⁡(π36)2)z_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right) 
       /6 ___    /pi\   6 ___   ___    /pi\\   6 ___    /pi\   6 ___   ___    /pi\
       |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
       |         \36/                  \36/|            \36/                  \36/
z6 = I*|------------- - -------------------| + ------------- + -------------------
       \      2                  2         /         2                  2
z6=263sin⁡(π36)2+26cos⁡(π36)2+i(−263cos⁡(π36)2+26sin⁡(π36)2)z_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right) 
z6 = 2^(1/6)*sqrt(3)*sin(pi/36)/2 + 2^(1/6)*cos(pi/36)/2 + i*(-2^(1/6)*sqrt(3)*cos(pi/36)/2 + 2^(1/6)*sin(pi/36)/2)
Respuesta numérica [src] 
z1 = -1.118190741335 - 0.0978290135264612*i
z2 = 0.474372959726412 + 1.01729609503589*i
z3 = 0.643817781608586 - 0.919467081509432*i
z4 = -0.474372959726412 - 1.01729609503589*i
z5 = 1.118190741335 + 0.0978290135264612*i
z6 = -0.643817781608586 + 0.919467081509432*i
z6 = -0.643817781608586 + 0.919467081509432*i
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