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z^5-sqrt3-i=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 5     ___        
z  - \/ 3  - I = 0
(z5−3)−i=0\left(z^{5} - \sqrt{3}\right) - i = 0
Simplificar
Solución detallada
Tenemos la ecuación
(z5−3)−i=0\left(z^{5} - \sqrt{3}\right) - i = 0
Ya que la potencia en la ecuación es igual a = 5 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 5 de las dos partes de la ecuación:
Obtenemos:
z55=3+i5\sqrt[5]{z^{5}} = \sqrt[5]{\sqrt{3} + i}
o
z=3+i5z = \sqrt[5]{\sqrt{3} + i}
Abrimos los paréntesis en el miembro derecho de la ecuación
z = i+sqrt+3)^1/5

Obtenemos la respuesta: z = (i + sqrt(3))^(1/5)

Las demás 5 raíces son complejas.
hacemos el cambio:
w=zw = z
entonces la ecuación será así:
w5=3+iw^{5} = \sqrt{3} + i
Cualquier número complejo se puede presentar que:
w=reipw = r e^{i p}
sustituimos en la ecuación
r5e5ip=3+ir^{5} e^{5 i p} = \sqrt{3} + i
donde
r=25r = \sqrt[5]{2}
- módulo del número complejo
Sustituyamos r:
e5ip=32+i2e^{5 i p} = \frac{\sqrt{3}}{2} + \frac{i}{2}
Usando la fórmula de Euler hallemos las raíces para p
isin⁡(5p)+cos⁡(5p)=32+i2i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = \frac{\sqrt{3}}{2} + \frac{i}{2}
es decir
cos⁡(5p)=32\cos{\left(5 p \right)} = \frac{\sqrt{3}}{2}
y
sin⁡(5p)=12\sin{\left(5 p \right)} = \frac{1}{2}
entonces
p=2πN5+π30p = \frac{2 \pi N}{5} + \frac{\pi}{30}
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=−2532+25i2w_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}
w2=2538+2558−582+15258−255i8−25i8+253i58−582w_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} i}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}
w3=−15258−25558−584−2558−584+2538−1525i58−584−253i58−584−25i8+255i8w_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{5} i}{8}
w4=−25558+588−2558+588−2558−588+25558−588+2538+25358−5858+582−1525i58+588−25i58−5858+582−253i58+588−253i58−588−25i8+1525i58−588w_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}
w5=−25358−5858+582−2558−588+2558+588+25558−588+2538+25558+588−253i58−588−25i8+253i58+588+25i58−5858+582+1525i58−588+1525i58+588w_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}
hacemos cambio inverso
w=zw = z
z=wz = w

Entonces la respuesta definitiva es:
z1=−2532+25i2z_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}
z2=2538+2558−582+15258−255i8−25i8+253i58−582z_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} i}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}
z3=−15258−25558−584−2558−584+2538−1525i58−584−253i58−584−25i8+255i8z_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{5} i}{8}
z4=−25558+588−2558+588−2558−588+25558−588+2538+25358−5858+582−1525i58+588−25i58−5858+582−253i58+588−253i58−588−25i8+1525i58−588z_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}
z5=−25358−5858+582−2558−588+2558+588+25558−588+2538+25558+588−253i58−588−25i8+253i58+588+25i58−5858+582+1525i58−588+1525i58+588z_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}
Gráfica
Respuesta rápida [src] 
       5 ___   5 ___   ___
     I*\/ 2    \/ 2 *\/ 3 
z1 = ------- - -----------
        2           2
z1=−2532+25i2z_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2} 
       /                                         ___________\              ___________                             
       |                                        /       ___ |             /       ___                              
       |                        5 ___   ___    /  5   \/ 5  |   5 ___    /  5   \/ 5                               
       |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - ----- |   \/ 2 *  /   - - -----    5 ___   ___   5 ___   ____
       |  \/ 2    \/ 2 *\/ 5                \/    8     8   |         \/    8     8      \/ 2 *\/ 3    \/ 2 *\/ 15 
z2 = I*|- ----- - ----------- + ----------------------------| + ---------------------- + ----------- + ------------
       \    8          8                     2              /             2                   8             8
z2=2538+2558−582+15258+i(−2558−258+25358−582)z_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{5}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) 
       /                                         ___________                     ___________\              ___________                                                 ___________
       |                                        /       ___                     /       ___ |             /       ___                                                 /       ___ 
       |                        5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5                                  5 ___   ___    /  5   \/ 5  
       |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 15 *  /   - - ----- |   \/ 2 *  /   - - -----    5 ___   ____   5 ___   ___   \/ 2 *\/ 5 *  /   - - ----- 
       |  \/ 2    \/ 2 *\/ 5                \/    8     8                   \/    8     8   |         \/    8     8      \/ 2 *\/ 15    \/ 2 *\/ 3                \/    8     8   
z3 = I*|- ----- + ----------- - ---------------------------- - -----------------------------| - ---------------------- - ------------ + ----------- - ----------------------------
       \    8          8                     4                               4              /             4                   8              8                     4
z3=−15258−25558−584−2558−584+2538+i(−152558−584−25358−584−258+2558)z_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{5}}{8}\right) 
       /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\              ___________              ___________                                  ___________                    ___________                    ___________      ___________
       |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |             /       ___              /       ___                                  /       ___                    /       ___                    /       ___      /       ___ 
       |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5     5 ___    /  5   \/ 5                   5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5      /  5   \/ 5  
       |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - + -----    \/ 2 *\/ 15 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + -----    5 ___   ___   \/ 2 *\/ 5 *  /   - + -----    \/ 2 *\/ 5 *  /   - - -----    \/ 2 *\/ 3 *  /   - - ----- *  /   - + ----- 
       |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |         \/    8     8            \/    8     8      \/ 2 *\/ 3                \/    8     8                  \/    8     8                  \/    8     8    \/    8     8   
z4 = I*|- ----- - --------------------------------------- - ---------------------------- - ---------------------------- - ----------------------------- + -----------------------------| - ---------------------- - ---------------------- + ----------- - ---------------------------- + ---------------------------- + ---------------------------------------------
       \    8                        2                                   8                              8                               8                               8              /             8                        8                   8                     8                              8                                       2
z4=−25558+588−2558+588−2558−588+25558−588+2538+25358−5858+582+i(−152558+588−2558−5858+582−25358+588−25358−588−258+152558−588)z_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}\right) 
       /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\              ___________                            ___________                    ___________                    ___________                    ___________      ___________
       |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |             /       ___                            /       ___                    /       ___                    /       ___                    /       ___      /       ___ 
       |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5                   5 ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5      /  5   \/ 5  
       |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - - -----    \/ 2 *\/ 15 *  /   - + ----- |   \/ 2 *  /   - - -----    5 ___   ___   \/ 2 *  /   - + -----    \/ 2 *\/ 5 *  /   - - -----    \/ 2 *\/ 5 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- *  /   - + ----- 
       |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |         \/    8     8      \/ 2 *\/ 3          \/    8     8                  \/    8     8                  \/    8     8                  \/    8     8    \/    8     8   
z5 = I*|- ----- + --------------------------------------- - ---------------------------- + ---------------------------- + ----------------------------- + -----------------------------| - ---------------------- + ----------- + ---------------------- + ---------------------------- + ---------------------------- - ---------------------------------------------
       \    8                        2                                   8                              8                               8                               8              /             8                   8                  8                           8                              8                                       2
z5=−25358−5858+582−2558−588+2558+588+25558−588+2538+25558+588+i(−25358−588−258+25358+588+2558−5858+582+152558−588+152558+588)z_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}\right) 
z5 = -2^(1/5)*sqrt(3)*sqrt(5/8 - sqrt(5)/8)*sqrt(sqrt(5)/8 + 5/8)/2 - 2^(1/5)*sqrt(5/8 - sqrt(5)/8)/8 + 2^(1/5)*sqrt(sqrt(5)/8 + 5/8)/8 + 2^(1/5)*sqrt(5)*sqrt(5/8 - sqrt(5)/8)/8 + 2^(1/5)*sqrt(3)/8 + 2^(1/5)*sqrt(5)*sqrt(sqrt(5)/8 + 5/8)/8 + i*(-2^(1/5)*sqrt(3)*sqrt(5/8 - sqrt(5)/8)/8 - 2^(1/5)/8 + 2^(1/5)*sqrt(3)*sqrt(sqrt(5)/8 + 5/8)/8 + 2^(1/5)*sqrt(5/8 - sqrt(5)/8)*sqrt(sqrt(5)/8 + 5/8)/2 + sqrt(15)*2^(1/5)*sqrt(5/8 - sqrt(5)/8)/8 + sqrt(15)*2^(1/5)*sqrt(sqrt(5)/8 + 5/8)/8)
Suma y producto de raíces [src] 
suma
                          /                                         ___________\              ___________                                  /                                         ___________                     ___________\              ___________                                                 ___________     /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\              ___________              ___________                                  ___________                    ___________                    ___________      ___________     /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\              ___________                            ___________                    ___________                    ___________                    ___________      ___________
                          |                                        /       ___ |             /       ___                                   |                                        /       ___                     /       ___ |             /       ___                                                 /       ___      |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |             /       ___              /       ___                                  /       ___                    /       ___                    /       ___      /       ___      |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |             /       ___                            /       ___                    /       ___                    /       ___                    /       ___      /       ___ 
                          |                        5 ___   ___    /  5   \/ 5  |   5 ___    /  5   \/ 5                                    |                        5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5                                  5 ___   ___    /  5   \/ 5       |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5     5 ___    /  5   \/ 5                   5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5      /  5   \/ 5       |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5                   5 ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5      /  5   \/ 5  
  5 ___   5 ___   ___     |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - ----- |   \/ 2 *  /   - - -----    5 ___   ___   5 ___   ____     |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 15 *  /   - - ----- |   \/ 2 *  /   - - -----    5 ___   ____   5 ___   ___   \/ 2 *\/ 5 *  /   - - -----      |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - + -----    \/ 2 *\/ 15 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + -----    5 ___   ___   \/ 2 *\/ 5 *  /   - + -----    \/ 2 *\/ 5 *  /   - - -----    \/ 2 *\/ 3 *  /   - - ----- *  /   - + -----      |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - - -----    \/ 2 *\/ 15 *  /   - + ----- |   \/ 2 *  /   - - -----    5 ___   ___   \/ 2 *  /   - + -----    \/ 2 *\/ 5 *  /   - - -----    \/ 2 *\/ 5 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- *  /   - + ----- 
I*\/ 2    \/ 2 *\/ 3      |  \/ 2    \/ 2 *\/ 5                \/    8     8   |         \/    8     8      \/ 2 *\/ 3    \/ 2 *\/ 15      |  \/ 2    \/ 2 *\/ 5                \/    8     8                   \/    8     8   |         \/    8     8      \/ 2 *\/ 15    \/ 2 *\/ 3                \/    8     8        |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |         \/    8     8            \/    8     8      \/ 2 *\/ 3                \/    8     8                  \/    8     8                  \/    8     8    \/    8     8        |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |         \/    8     8      \/ 2 *\/ 3          \/    8     8                  \/    8     8                  \/    8     8                  \/    8     8    \/    8     8   
------- - ----------- + I*|- ----- - ----------- + ----------------------------| + ---------------------- + ----------- + ------------ + I*|- ----- + ----------- - ---------------------------- - -----------------------------| - ---------------------- - ------------ + ----------- - ---------------------------- + I*|- ----- - --------------------------------------- - ---------------------------- - ---------------------------- - ----------------------------- + -----------------------------| - ---------------------- - ---------------------- + ----------- - ---------------------------- + ---------------------------- + --------------------------------------------- + I*|- ----- + --------------------------------------- - ---------------------------- + ---------------------------- + ----------------------------- + -----------------------------| - ---------------------- + ----------- + ---------------------- + ---------------------------- + ---------------------------- - ---------------------------------------------
   2           2          \    8          8                     2              /             2                   8             8           \    8          8                     4                               4              /             4                   8              8                     4                   \    8                        2                                   8                              8                               8                               8              /             8                        8                   8                     8                              8                                       2                           \    8                        2                                   8                              8                               8                               8              /             8                   8                  8                           8                              8                                       2
((−25558+588−2558+588−2558−588+25558−588+2538+25358−5858+582+i(−152558+588−2558−5858+582−25358+588−25358−588−258+152558−588))+((−15258−25558−584−2558−584+2538+i(−152558−584−25358−584−258+2558))+((2538+2558−582+15258+i(−2558−258+25358−582))+(−2532+25i2))))+(−25358−5858+582−2558−588+2558+588+25558−588+2538+25558+588+i(−25358−588−258+25358+588+2558−5858+582+152558−588+152558+588))\left(\left(- \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}\right)\right) + \left(\left(- \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{5}}{8}\right)\right) + \left(\left(\frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{5}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right) + \left(- \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}\right)\right)\right)\right) + \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}\right)\right) 
=
  /                                         ___________\     /                                         ___________                     ___________\     /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\     /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\          
  |                                        /       ___ |     |                                        /       ___                     /       ___ |     |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |     |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |          
  |                        5 ___   ___    /  5   \/ 5  |     |                        5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |     |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |     |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |          
  |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - ----- |     |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 15 *  /   - - ----- |     |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - - -----    \/ 2 *\/ 15 *  /   - + ----- |     |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - + -----    \/ 2 *\/ 15 *  /   - - ----- |     5 ___
  |  \/ 2    \/ 2 *\/ 5                \/    8     8   |     |  \/ 2    \/ 2 *\/ 5                \/    8     8                   \/    8     8   |     |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |     |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |   I*\/ 2 
I*|- ----- - ----------- + ----------------------------| + I*|- ----- + ----------- - ---------------------------- - -----------------------------| + I*|- ----- + --------------------------------------- - ---------------------------- + ---------------------------- + ----------------------------- + -----------------------------| + I*|- ----- - --------------------------------------- - ---------------------------- - ---------------------------- - ----------------------------- + -----------------------------| + -------
  \    8          8                     2              /     \    8          8                     4                               4              /     \    8                        2                                   8                              8                               8                               8              /     \    8                        2                                   8                              8                               8                               8              /      2
i(−152558+588−2558−5858+582−25358+588−25358−588−258+152558−588)+i(−152558−584−25358−584−258+2558)+i(−2558−258+25358−582)+25i2+i(−25358−588−258+25358+588+2558−5858+582+152558−588+152558+588)i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}\right) + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{5}}{8}\right) + i \left(- \frac{\sqrt[5]{2} \sqrt{5}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \frac{\sqrt[5]{2} i}{2} + i \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}\right) 
producto
                        /  /                                         ___________\              ___________                             \ /  /                                         ___________                     ___________\              ___________                                                 ___________\ /  /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\              ___________              ___________                                  ___________                    ___________                    ___________      ___________\ /  /                     ___________      ___________                    ___________                    ___________                     ___________                     ___________\              ___________                            ___________                    ___________                    ___________                    ___________      ___________\
                        |  |                                        /       ___ |             /       ___                              | |  |                                        /       ___                     /       ___ |             /       ___                                                 /       ___ | |  |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |             /       ___              /       ___                                  /       ___                    /       ___                    /       ___      /       ___ | |  |                    /       ___      /       ___                    /       ___                    /       ___                     /       ___                     /       ___ |             /       ___                            /       ___                    /       ___                    /       ___                    /       ___      /       ___ |
                        |  |                        5 ___   ___    /  5   \/ 5  |   5 ___    /  5   \/ 5                               | |  |                        5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5                                  5 ___   ___    /  5   \/ 5  | |  |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5     5 ___    /  5   \/ 5                   5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5      /  5   \/ 5  | |  |          5 ___    /  5   \/ 5      /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ____    /  5   \/ 5     5 ___   ____    /  5   \/ 5  |   5 ___    /  5   \/ 5                   5 ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5     5 ___   ___    /  5   \/ 5      /  5   \/ 5  |
/  5 ___   5 ___   ___\ |  |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - ----- |   \/ 2 *  /   - - -----    5 ___   ___   5 ___   ____| |  |  5 ___   5 ___   ___   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 15 *  /   - - ----- |   \/ 2 *  /   - - -----    5 ___   ____   5 ___   ___   \/ 2 *\/ 5 *  /   - - ----- | |  |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - + -----    \/ 2 *\/ 15 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + -----    5 ___   ___   \/ 2 *\/ 5 *  /   - + -----    \/ 2 *\/ 5 *  /   - - -----    \/ 2 *\/ 3 *  /   - - ----- *  /   - + ----- | |  |  5 ___   \/ 2 *  /   - - ----- *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 15 *  /   - - -----    \/ 2 *\/ 15 *  /   - + ----- |   \/ 2 *  /   - - -----    5 ___   ___   \/ 2 *  /   - + -----    \/ 2 *\/ 5 *  /   - - -----    \/ 2 *\/ 5 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- *  /   - + ----- |
|I*\/ 2    \/ 2 *\/ 3 | |  |  \/ 2    \/ 2 *\/ 5                \/    8     8   |         \/    8     8      \/ 2 *\/ 3    \/ 2 *\/ 15 | |  |  \/ 2    \/ 2 *\/ 5                \/    8     8                   \/    8     8   |         \/    8     8      \/ 2 *\/ 15    \/ 2 *\/ 3                \/    8     8   | |  |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |         \/    8     8            \/    8     8      \/ 2 *\/ 3                \/    8     8                  \/    8     8                  \/    8     8    \/    8     8   | |  |  \/ 2          \/    8     8    \/    8     8                  \/    8     8                  \/    8     8                   \/    8     8                   \/    8     8   |         \/    8     8      \/ 2 *\/ 3          \/    8     8                  \/    8     8                  \/    8     8                  \/    8     8    \/    8     8   |
|------- - -----------|*|I*|- ----- - ----------- + ----------------------------| + ---------------------- + ----------- + ------------|*|I*|- ----- + ----------- - ---------------------------- - -----------------------------| - ---------------------- - ------------ + ----------- - ----------------------------|*|I*|- ----- - --------------------------------------- - ---------------------------- - ---------------------------- - ----------------------------- + -----------------------------| - ---------------------- - ---------------------- + ----------- - ---------------------------- + ---------------------------- + ---------------------------------------------|*|I*|- ----- + --------------------------------------- - ---------------------------- + ---------------------------- + ----------------------------- + -----------------------------| - ---------------------- + ----------- + ---------------------- + ---------------------------- + ---------------------------- - ---------------------------------------------|
\   2           2     / \  \    8          8                     2              /             2                   8             8      / \  \    8          8                     4                               4              /             4                   8              8                     4              / \  \    8                        2                                   8                              8                               8                               8              /             8                        8                   8                     8                              8                                       2                      / \  \    8                        2                                   8                              8                               8                               8              /             8                   8                  8                           8                              8                                       2                      /
(−2532+25i2)(2538+2558−582+15258+i(−2558−258+25358−582))(−15258−25558−584−2558−584+2538+i(−152558−584−25358−584−258+2558))(−25558+588−2558+588−2558−588+25558−588+2538+25358−5858+582+i(−152558+588−2558−5858+582−25358+588−25358−588−258+152558−588))(−25358−5858+582−2558−588+2558+588+25558−588+2538+25558+588+i(−25358−588−258+25358+588+2558−5858+582+152558−588+152558+588))\left(- \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}\right) \left(\frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{5}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right) \left(- \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{5}}{8}\right)\right) \left(- \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}\right)\right) \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}\right)\right) 
=
                 _______________      _______________      _________________      _________________      _______________        ______________
                /           ___      /           ___      /             ___      /             ___      /           ___        /          ___ 
      ___   5*\/  -30 + 6*\/ 5     \/  50 + 10*\/ 5     \/  -150 + 30*\/ 5     \/  -150 - 30*\/ 5     \/  50 - 10*\/ 5     5*\/  10 - 2*\/ 5  
I + \/ 3  - -------------------- - ------------------ - -------------------- + -------------------- + ------------------ + -------------------
                     64                    32                    64                     32                    64                    64
−105+5032+50−10564+510−2564+3−5−30+6564−−150+30564+−150−30532+i- \frac{\sqrt{10 \sqrt{5} + 50}}{32} + \frac{\sqrt{50 - 10 \sqrt{5}}}{64} + \frac{5 \sqrt{10 - 2 \sqrt{5}}}{64} + \sqrt{3} - \frac{5 \sqrt{-30 + 6 \sqrt{5}}}{64} - \frac{\sqrt{-150 + 30 \sqrt{5}}}{64} + \frac{\sqrt{-150 - 30 \sqrt{5}}}{32} + i 
i + sqrt(3) - 5*sqrt(-30 + 6*sqrt(5))/64 - sqrt(50 + 10*sqrt(5))/32 - sqrt(-150 + 30*sqrt(5))/64 + sqrt(-150 - 30*sqrt(5))/32 + sqrt(50 - 10*sqrt(5))/64 + 5*sqrt(10 - 2*sqrt(5))/64
Respuesta numérica [src] 
z1 = 0.238827817227012 + 1.12359653990722*i
z2 = -0.853649238550441 - 0.768629226802589*i
z3 = 1.14240566521807 + 0.120071673805922*i
z4 = -0.994801956712828 + 0.574349177498518*i
z5 = 0.467217712818188 - 1.04938816440907*i
z5 = 0.467217712818188 - 1.04938816440907*i
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