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z^5-sqrt3-i=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$\left(z^{5} - \sqrt{3}\right) - i = 0$$
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:
$$\sqrt[5]{z^{5}} = \sqrt[5]{\sqrt{3} + i}$$
o
$$z = \sqrt[5]{\sqrt{3} + i}$$
Abrimos los paréntesis en el miembro derecho de la ecuación
Obtenemos la respuesta: z = (i + sqrt(3))^(1/5)
Las demás 5 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{5} = \sqrt{3} + i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{5} e^{5 i p} = \sqrt{3} + i$$
donde
$$r = \sqrt[5]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{5 i p} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
es decir
$$\cos{\left(5 p \right)} = \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(5 p \right)} = \frac{1}{2}$$
entonces
$$p = \frac{2 \pi N}{5} + \frac{\pi}{30}$$
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:
$$w_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}$$
$$w_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} i}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$w_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{5} i}{8}$$
$$w_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}$$
$$w_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}$$
$$z_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} i}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$z_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{5} i}{8}$$
$$z_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}$$
$$z_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}$$
$$\left(z^{5} - \sqrt{3}\right) - i = 0$$
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:
$$\sqrt[5]{z^{5}} = \sqrt[5]{\sqrt{3} + i}$$
o
$$z = \sqrt[5]{\sqrt{3} + i}$$
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:
$$w = z$$
entonces la ecuación será así:
$$w^{5} = \sqrt{3} + i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{5} e^{5 i p} = \sqrt{3} + i$$
donde
$$r = \sqrt[5]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{5 i p} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
es decir
$$\cos{\left(5 p \right)} = \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(5 p \right)} = \frac{1}{2}$$
entonces
$$p = \frac{2 \pi N}{5} + \frac{\pi}{30}$$
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:
$$w_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}$$
$$w_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} i}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$w_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{5} i}{8}$$
$$w_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}$$
$$w_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}$$
$$z_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} i}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$z_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{5} i}{8}$$
$$z_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}$$
$$z_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2} i}{8} + \frac{\sqrt[5]{2} \sqrt{3} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}$$
Gráfica
Respuesta rápida
[src]
5 ___ 5 ___ ___
I*\/ 2 \/ 2 *\/ 3
z1 = ------- - -----------
2 2
$$z_{1} = - \frac{\sqrt[5]{2} \sqrt{3}}{2} + \frac{\sqrt[5]{2} i}{2}$$
/ ___________\ ___________
| / ___ | / ___
| 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
$$z_{2} = \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{5}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)$$
/ ___________ ___________\ ___________ ___________
| / ___ / ___ | / ___ / ___
| 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
$$z_{3} = - \frac{\sqrt{15} \sqrt[5]{2}}{8} - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{5}}{8}\right)$$
/ ___________ ___________ ___________ ___________ ___________ ___________\ ___________ ___________ ___________ ___________ ___________ ___________
| / ___ / ___ / ___ / ___ / ___ / ___ | / ___ / ___ / ___ / ___ / ___ / ___
| 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
$$z_{4} = - \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}\right)$$
/ ___________ ___________ ___________ ___________ ___________ ___________\ ___________ ___________ ___________ ___________ ___________ ___________
| / ___ / ___ / ___ / ___ / ___ / ___ | / ___ / ___ / ___ / ___ / ___ / ___
| 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
$$z_{5} = - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{3}}{8} + \frac{\sqrt[5]{2} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + i \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}\right)$$
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
$$\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
$$i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8}\right) + i \left(- \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{5}}{8}\right) + i \left(- \frac{\sqrt[5]{2} \sqrt{5}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \frac{\sqrt[5]{2} i}{2} + i \left(- \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{\sqrt[5]{2}}{8} + \frac{\sqrt[5]{2} \sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{\sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{\sqrt{15} \sqrt[5]{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8}\right)$$
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 /
$$\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
$$- \frac{\sqrt{10 \sqrt{5} + 50}}{32} + \frac{\sqrt{50 - 10 \sqrt{5}}}{64} + \frac{5 \sqrt{10 - 2 \sqrt{5}}}{64} + \sqrt{3} - \frac{5 \sqrt{-30 + 6 \sqrt{5}}}{64} - \frac{\sqrt{-150 + 30 \sqrt{5}}}{64} + \frac{\sqrt{-150 - 30 \sqrt{5}}}{32} + i$$
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