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- La ecuación:
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Ecuación 3^x=2
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Ecuación 8*x=2
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- Expresar {x} en función de y en la ecuación:
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- -11*x+9*y=-20
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Expresiones idénticas
- z^ cuatro +(2i/(uno +√(tres)i))= cero
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- z en el grado cuatro más (2i dividir por (uno más √(tres)i)) es igual a cero
- z4+(2i/(1+√(3)i))=0
- z4+2i/1+√3i=0
- z⁴+(2i/(1+√(3)i))=0
- z^4+2i/1+√3i=0
- z^4+(2i/(1+√(3)i))=O
- z^4+(2i dividir por (1+√(3)i))=0
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Expresiones semejantes
- z^4+(2i/(1-√(3)i))=0
- z^4-(2i/(1+√(3)i))=0
z^4+(2i/(1+√(3)i))=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
4 2*I
z + ----------- = 0
___
1 + \/ 3 *I
$$z^{4} + \frac{2 i}{1 + \sqrt{3} i} = 0$$
Solución detallada
Tenemos la ecuación
$$z^{4} + \frac{2 i}{1 + \sqrt{3} i} = 0$$
Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -2*i/(1 + i*sqrt(3)) complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 4 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{4} = - \frac{2 i}{1 + \sqrt{3} i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{4} e^{4 i p} = - \frac{2 i}{1 + \sqrt{3} i}$$
donde
$$r = 1$$
- módulo del número complejo
Sustituyamos r:
$$e^{4 i p} = - \frac{2 i}{1 + \sqrt{3} i}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = - \frac{2 i}{1 + \sqrt{3} i}$$
es decir
$$\cos{\left(4 p \right)} = - \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(4 p \right)} = - \frac{1}{2}$$
entonces
$$p = \frac{\pi N}{2} + \frac{\pi}{24}$$
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:
$$w_{1} = \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$w_{2} = - \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$w_{3} = - \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$w_{4} = - \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{2} = - \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{3} = - \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{4} = - \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z^{4} + \frac{2 i}{1 + \sqrt{3} i} = 0$$
Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -2*i/(1 + i*sqrt(3)) complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 4 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{4} = - \frac{2 i}{1 + \sqrt{3} i}$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{4} e^{4 i p} = - \frac{2 i}{1 + \sqrt{3} i}$$
donde
$$r = 1$$
- módulo del número complejo
Sustituyamos r:
$$e^{4 i p} = - \frac{2 i}{1 + \sqrt{3} i}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = - \frac{2 i}{1 + \sqrt{3} i}$$
es decir
$$\cos{\left(4 p \right)} = - \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(4 p \right)} = - \frac{1}{2}$$
entonces
$$p = \frac{\pi N}{2} + \frac{\pi}{24}$$
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:
$$w_{1} = \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$w_{2} = - \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$w_{3} = - \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$w_{4} = - \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{2} = - \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{3} = - \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{4} = - \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
Gráfica
Respuesta rápida
[src]
___________ / ___________ ___________\ ___________
/ ___ | / ___ / ___ | / ___
/ 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2
/ - + ----- | / - - ----- \/ 3 * / - + ----- | \/ 3 * / - - -----
\/ 2 4 |\/ 2 4 \/ 2 4 | \/ 2 4
z1 = ---------------- + I*|---------------- - ----------------------| + ----------------------
2 \ 2 2 / 2
$$z_{1} = \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$
___________ / ___________ ___________\ ___________
/ ___ | / ___ / ___ | / ___
/ 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2
/ - - ----- | / - + ----- \/ 3 * / - - ----- | \/ 3 * / - + -----
\/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4
z2 = ---------------- + I*|- ---------------- - ----------------------| - ----------------------
2 \ 2 2 / 2
$$z_{2} = - \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$
___________ / ___________ ___________\ ___________
/ ___ | / ___ / ___ | / ___
/ 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2
/ - - ----- | / - + ----- \/ 3 * / - - ----- | \/ 3 * / - + -----
\/ 2 4 |\/ 2 4 \/ 2 4 | \/ 2 4
z3 = - ---------------- + I*|---------------- + ----------------------| + ----------------------
2 \ 2 2 / 2
$$z_{3} = - \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$
___________ / ___________ ___________\ ___________
/ ___ | / ___ / ___ | / ___
/ 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2
/ - + ----- | / - - ----- \/ 3 * / - + ----- | \/ 3 * / - - -----
\/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4
z4 = - ---------------- + I*|- ---------------- + ----------------------| - ----------------------
2 \ 2 2 / 2
$$z_{4} = - \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$
z4 = -sqrt(sqrt(2)/4 + 1/2)/2 - sqrt(3)*sqrt(1/2 - sqrt(2)/4)/2 + i*(-sqrt(1/2 - sqrt(2)/4)/2 + sqrt(3)*sqrt(sqrt(2)/4 + 1/2)/2)
Suma y producto de raíces
[src]
suma
___________ / ___________ ___________\ ___________ ___________ / ___________ ___________\ ___________ ___________ / ___________ ___________\ ___________ ___________ / ___________ ___________\ ___________
/ ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___ / ___ | / ___
/ 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 / 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 / 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 / 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2
/ - + ----- | / - - ----- \/ 3 * / - + ----- | \/ 3 * / - - ----- / - - ----- | / - + ----- \/ 3 * / - - ----- | \/ 3 * / - + ----- / - - ----- | / - + ----- \/ 3 * / - - ----- | \/ 3 * / - + ----- / - + ----- | / - - ----- \/ 3 * / - + ----- | \/ 3 * / - - -----
\/ 2 4 |\/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 |\/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4
---------------- + I*|---------------- - ----------------------| + ---------------------- + ---------------- + I*|- ---------------- - ----------------------| - ---------------------- + - ---------------- + I*|---------------- + ----------------------| + ---------------------- + - ---------------- + I*|- ---------------- + ----------------------| - ----------------------
2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(\left(\left(- \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)\right) + \left(- \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)\right) + \left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)$$
=
/ ___________ ___________\ / ___________ ___________\ / ___________ ___________\ / ___________ ___________\ | / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | / 1 \/ 2 ___ / 1 \/ 2 | | / 1 \/ 2 ___ / 1 \/ 2 | | / 1 \/ 2 ___ / 1 \/ 2 | | / 1 \/ 2 ___ / 1 \/ 2 | | / - - ----- \/ 3 * / - + ----- | | / - + ----- \/ 3 * / - - ----- | | / - - ----- \/ 3 * / - + ----- | | / - + ----- \/ 3 * / - - ----- | |\/ 2 4 \/ 2 4 | |\/ 2 4 \/ 2 4 | | \/ 2 4 \/ 2 4 | | \/ 2 4 \/ 2 4 | I*|---------------- - ----------------------| + I*|---------------- + ----------------------| + I*|- ---------------- + ----------------------| + I*|- ---------------- - ----------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) + i \left(\frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$
producto
/ ___________ / ___________ ___________\ ___________\ / ___________ / ___________ ___________\ ___________\ / ___________ / ___________ ___________\ ___________\ / ___________ / ___________ ___________\ ___________\ | / ___ | / ___ / ___ | / ___ | | / ___ | / ___ / ___ | / ___ | | / ___ | / ___ / ___ | / ___ | | / ___ | / ___ / ___ | / ___ | | / 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 | | / 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 | | / 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 | | / 1 \/ 2 | / 1 \/ 2 ___ / 1 \/ 2 | ___ / 1 \/ 2 | | / - + ----- | / - - ----- \/ 3 * / - + ----- | \/ 3 * / - - ----- | | / - - ----- | / - + ----- \/ 3 * / - - ----- | \/ 3 * / - + ----- | | / - - ----- | / - + ----- \/ 3 * / - - ----- | \/ 3 * / - + ----- | | / - + ----- | / - - ----- \/ 3 * / - + ----- | \/ 3 * / - - ----- | |\/ 2 4 |\/ 2 4 \/ 2 4 | \/ 2 4 | |\/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 | | \/ 2 4 |\/ 2 4 \/ 2 4 | \/ 2 4 | | \/ 2 4 | \/ 2 4 \/ 2 4 | \/ 2 4 | |---------------- + I*|---------------- - ----------------------| + ----------------------|*|---------------- + I*|- ---------------- - ----------------------| - ----------------------|*|- ---------------- + I*|---------------- + ----------------------| + ----------------------|*|- ---------------- + I*|- ---------------- + ----------------------| - ----------------------| \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(\frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)$$
=
___ I \/ 3 - + ----- 2 2
$$\frac{\sqrt{3}}{2} + \frac{i}{2}$$
i/2 + sqrt(3)/2
Respuesta numérica
[src]
z1 = 0.608761429008721 + 0.793353340291235*i
z2 = -0.793353340291235 + 0.608761429008721*i
z3 = 0.793353340291235 - 0.608761429008721*i
z4 = -0.608761429008721 - 0.793353340291235*i
z4 = -0.608761429008721 - 0.793353340291235*i