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La ecuación:
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Expresar {x} en función de y en la ecuación:
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-14*x-12*y=5
Expresiones idénticas
z^ doce =sqrt(tres)-i
z en el grado 12 es igual a raíz cuadrada de (3) menos i
z en el grado doce es igual a raíz cuadrada de (tres) menos i
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z12=sqrt(3)-i
z12=sqrt3-i
z^12=sqrt3-i
Expresiones semejantes
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Expresiones con funciones
Raíz cuadrada sqrt
sqrt(x+4)=x-3
sqrt(x-3)=5-x
sqrt(2*x+3)=x
sqrt(x^2-x-3)=3
sqrt(x)=-2

z^12=sqrt(3)-i la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

 12     ___    
z   = \/ 3  - I

z^{12} = \sqrt{3} - i

Simplificar

Solución detallada

Tenemos la ecuación

z^{12} = \sqrt{3} - i

Ya que la potencia en la ecuación es igual a = 12 y miembro libre = sqrt(3) - i complejo,
significa que la ecuación correspondiente no tiene soluciones reales

Las demás 12 raíces son complejas.
hacemos el cambio:

w = z

entonces la ecuación será así:

w^{12} = \sqrt{3} - i

Cualquier número complejo se puede presentar que:

w = r e^{i p}

sustituimos en la ecuación

r^{12} e^{12 i p} = \sqrt{3} - i

donde

r = \sqrt[12]{2}

- módulo del número complejo
Sustituyamos r:

e^{12 i p} = \frac{\sqrt{3}}{2} - \frac{i}{2}

Usando la fórmula de Euler hallemos las raíces para p

i \sin{\left(12 p \right)} + \cos{\left(12 p \right)} = \frac{\sqrt{3}}{2} - \frac{i}{2}

es decir

\cos{\left(12 p \right)} = \frac{\sqrt{3}}{2}

\sin{\left(12 p \right)} = - \frac{1}{2}

entonces

p = \frac{\pi N}{6} - \frac{\pi}{72}

donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para w
Es decir, la solución será para w:

w_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}

w_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}

w_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}

w_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}

w_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}

w_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}

w_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}

w_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}

w_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}

w_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}

w_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}

w_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}

hacemos cambio inverso

w = z

z = w

Entonces la respuesta definitiva es:

z_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}

z_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}

z_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}

z_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}

z_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}

z_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}

z_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}

z_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}

z_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}

z_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}

z_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}

z_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}

Gráfica

Respuesta rápida [src]

       12___    /pi\     12___    /pi\
z1 = - \/ 2 *sin|--| - I*\/ 2 *cos|--|
                \72/              \72/

z_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}

     12___    /pi\     12___    /pi\
z2 = \/ 2 *sin|--| + I*\/ 2 *cos|--|
              \72/              \72/

z_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}

       12___    /pi\     12___    /pi\
z3 = - \/ 2 *cos|--| + I*\/ 2 *sin|--|
                \72/              \72/

z_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}

     12___    /pi\     12___    /pi\
z4 = \/ 2 *cos|--| - I*\/ 2 *sin|--|
              \72/              \72/

z_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}

       /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
       |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|
       |           \72/                  \72/|            \72/                  \72/
z5 = I*|- ------------- - -------------------| - ------------- + -------------------
       \        2                  2         /         2                  2

z_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)

       /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
       |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|
       |         \72/                  \72/|            \72/                  \72/
z6 = I*|------------- - -------------------| + ------------- + -------------------
       \      2                  2         /         2                  2

z_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)

       /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
       |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
       |         \72/                  \72/|            \72/                  \72/
z7 = I*|------------- - -------------------| - ------------- - -------------------
       \      2                  2         /         2                  2

z_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)

       /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
       |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
       |         \72/                  \72/|            \72/                  \72/
z8 = I*|------------- + -------------------| - ------------- + -------------------
       \      2                  2         /         2                  2

z_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)

       /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
       |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
       |           \72/                  \72/|            \72/                  \72/
z9 = I*|- ------------- - -------------------| + ------------- - -------------------
       \        2                  2         /         2                  2

z_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)

        /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
        |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|
        |           \72/                  \72/|            \72/                  \72/
z10 = I*|- ------------- + -------------------| + ------------- + -------------------
        \        2                  2         /         2                  2

z_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)

        /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
        |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|
        |           \72/                  \72/|            \72/                  \72/
z11 = I*|- ------------- + -------------------| - ------------- - -------------------
        \        2                  2         /         2                  2

z_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)

        /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
        |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|
        |         \72/                  \72/|            \72/                  \72/
z12 = I*|------------- + -------------------| + ------------- - -------------------
        \      2                  2         /         2                  2

z_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)

z12 = -2^(1/12)*sqrt(3)*cos(pi/72)/2 + 2^(1/12)*sin(pi/72)/2 + i*(2^(1/12)*sqrt(3)*sin(pi/72)/2 + 2^(1/12)*cos(pi/72)/2)

Suma y producto de raíces [src]

suma

                                                                                                                                              /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\     /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\     /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\     /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\     /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\     /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\     /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\     /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\
                                                                                                                                              |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|     |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|     |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|     |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|     |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|     |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|     |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|     |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|
  12___    /pi\     12___    /pi\   12___    /pi\     12___    /pi\     12___    /pi\     12___    /pi\   12___    /pi\     12___    /pi\     |           \72/                  \72/|            \72/                  \72/     |         \72/                  \72/|            \72/                  \72/     |         \72/                  \72/|            \72/                  \72/     |         \72/                  \72/|            \72/                  \72/     |           \72/                  \72/|            \72/                  \72/     |           \72/                  \72/|            \72/                  \72/     |           \72/                  \72/|            \72/                  \72/     |         \72/                  \72/|            \72/                  \72/
- \/ 2 *sin|--| - I*\/ 2 *cos|--| + \/ 2 *sin|--| + I*\/ 2 *cos|--| + - \/ 2 *cos|--| + I*\/ 2 *sin|--| + \/ 2 *cos|--| - I*\/ 2 *sin|--| + I*|- ------------- - -------------------| - ------------- + ------------------- + I*|------------- - -------------------| + ------------- + ------------------- + I*|------------- - -------------------| - ------------- - ------------------- + I*|------------- + -------------------| - ------------- + ------------------- + I*|- ------------- - -------------------| + ------------- - ------------------- + I*|- ------------- + -------------------| + ------------- + ------------------- + I*|- ------------- + -------------------| - ------------- - ------------------- + I*|------------- + -------------------| + ------------- - -------------------
           \72/              \72/            \72/              \72/              \72/              \72/            \72/              \72/     \        2                  2         /         2                  2              \      2                  2         /         2                  2              \      2                  2         /         2                  2              \      2                  2         /         2                  2              \        2                  2         /         2                  2              \        2                  2         /         2                  2              \        2                  2         /         2                  2              \      2                  2         /         2                  2

\left(\left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right) + \left(\left(\left(- \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right) + \left(\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right)\right) + \left(- \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right)\right)\right)\right) + \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)\right)\right) + \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)\right)\right) + \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)\right)\right) + \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)

  /12___    /pi\   12___   ___    /pi\\     /12___    /pi\   12___   ___    /pi\\     /12___    /pi\   12___   ___    /pi\\     /12___    /pi\   12___   ___    /pi\\     /  12___    /pi\   12___   ___    /pi\\     /  12___    /pi\   12___   ___    /pi\\     /  12___    /pi\   12___   ___    /pi\\     /  12___    /pi\   12___   ___    /pi\\
  |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||     |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||     |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||     |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||     |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||     |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||     |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||     |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||
  |         \72/                  \72/|     |         \72/                  \72/|     |         \72/                  \72/|     |         \72/                  \72/|     |           \72/                  \72/|     |           \72/                  \72/|     |           \72/                  \72/|     |           \72/                  \72/|
I*|------------- + -------------------| + I*|------------- - -------------------| + I*|------------- + -------------------| + I*|------------- - -------------------| + I*|- ------------- + -------------------| + I*|- ------------- - -------------------| + I*|- ------------- + -------------------| + I*|- ------------- - -------------------|
  \      2                  2         /     \      2                  2         /     \      2                  2         /     \      2                  2         /     \        2                  2         /     \        2                  2         /     \        2                  2         /     \        2                  2         /

i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)

producto

                                                                                                                                            /  /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\ /  /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\ /  /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\ /  /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\ /  /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\ /  /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\ /  /  12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\ /  /12___    /pi\   12___   ___    /pi\\   12___    /pi\   12___   ___    /pi\\
                                                                                                                                            |  |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|| |  |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|| |  |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|| |  |\/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|| |  |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|| |  |  \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||   \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--|| |  |  \/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--|| |  |\/ 2 *cos|--|   \/ 2 *\/ 3 *sin|--||   \/ 2 *sin|--|   \/ 2 *\/ 3 *cos|--||
/  12___    /pi\     12___    /pi\\ /12___    /pi\     12___    /pi\\ /  12___    /pi\     12___    /pi\\ /12___    /pi\     12___    /pi\\ |  |           \72/                  \72/|            \72/                  \72/| |  |         \72/                  \72/|            \72/                  \72/| |  |         \72/                  \72/|            \72/                  \72/| |  |         \72/                  \72/|            \72/                  \72/| |  |           \72/                  \72/|            \72/                  \72/| |  |           \72/                  \72/|            \72/                  \72/| |  |           \72/                  \72/|            \72/                  \72/| |  |         \72/                  \72/|            \72/                  \72/|
|- \/ 2 *sin|--| - I*\/ 2 *cos|--||*|\/ 2 *sin|--| + I*\/ 2 *cos|--||*|- \/ 2 *cos|--| + I*\/ 2 *sin|--||*|\/ 2 *cos|--| - I*\/ 2 *sin|--||*|I*|- ------------- - -------------------| - ------------- + -------------------|*|I*|------------- - -------------------| + ------------- + -------------------|*|I*|------------- - -------------------| - ------------- - -------------------|*|I*|------------- + -------------------| - ------------- + -------------------|*|I*|- ------------- - -------------------| + ------------- - -------------------|*|I*|- ------------- + -------------------| + ------------- + -------------------|*|I*|- ------------- + -------------------| - ------------- - -------------------|*|I*|------------- + -------------------| + ------------- - -------------------|
\           \72/              \72// \         \72/              \72// \           \72/              \72// \         \72/              \72// \  \        2                  2         /         2                  2         / \  \      2                  2         /         2                  2         / \  \      2                  2         /         2                  2         / \  \      2                  2         /         2                  2         / \  \        2                  2         /         2                  2         / \  \        2                  2         /         2                  2         / \  \        2                  2         /         2                  2         / \  \      2                  2         /         2                  2         /

\left(- \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right) \left(\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right) \left(- \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right) \left(\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right) \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)

                                                                                                                                                  -pi*I 
                               2                              2                      2                            2                            2  ------
  /     /11*pi\        /11*pi\\  /     /23*pi\        /13*pi\\  /     /pi\      /pi\\  /     /13*pi\      /13*pi\\  /     /11*pi\      /25*pi\\     36  
2*|- sin|-----| + I*cos|-----|| *|- sin|-----| + I*sin|-----|| *|I*cos|--| + sin|--|| *|I*cos|-----| + sin|-----|| *|I*sin|-----| + sin|-----|| *e      
  \     \  72 /        \  72 //  \     \  72 /        \  72 //  \     \72/      \72//  \     \  72 /      \  72 //  \     \  72 /      \  72 //

2 \left(\sin{\left(\frac{\pi}{72} \right)} + i \cos{\left(\frac{\pi}{72} \right)}\right)^{2} \left(- \sin{\left(\frac{11 \pi}{72} \right)} + i \cos{\left(\frac{11 \pi}{72} \right)}\right)^{2} \left(\sin{\left(\frac{13 \pi}{72} \right)} + i \cos{\left(\frac{13 \pi}{72} \right)}\right)^{2} \left(- \sin{\left(\frac{23 \pi}{72} \right)} + i \sin{\left(\frac{13 \pi}{72} \right)}\right)^{2} \left(\sin{\left(\frac{25 \pi}{72} \right)} + i \sin{\left(\frac{11 \pi}{72} \right)}\right)^{2} e^{- \frac{i \pi}{36}}

2*(-sin(11*pi/72) + i*cos(11*pi/72))^2*(-sin(23*pi/72) + i*sin(13*pi/72))^2*(i*cos(pi/72) + sin(pi/72))^2*(i*cos(13*pi/72) + sin(13*pi/72))^2*(i*sin(11*pi/72) + sin(25*pi/72))^2*exp(-pi*i/36)

Respuesta numérica [src]

z1 = 0.0462131311121356 + 1.05845472025127*i

z2 = -0.489205614594103 + 0.939755242049215*i

z3 = -0.569249105657163 - 0.89354211093708*i

z4 = -0.939755242049215 - 0.489205614594103*i

z5 = 0.569249105657163 + 0.89354211093708*i

z6 = -0.89354211093708 + 0.569249105657163*i

z7 = -0.0462131311121356 - 1.05845472025127*i

z8 = -1.05845472025127 + 0.0462131311121356*i

z9 = 0.939755242049215 + 0.489205614594103*i

z10 = 0.89354211093708 - 0.569249105657163*i

z11 = 0.489205614594103 - 0.939755242049215*i

z12 = 1.05845472025127 - 0.0462131311121356*i

z12 = 1.05845472025127 - 0.0462131311121356*i

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z^12=sqrt(3)-i la ecuación

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