x^8=-3 (уравнение)

Найду корень уравнения: x^8=-3

Вы ввели

 8     
x  = -3

$$x^{8} = -3$$

Подробное решение

Дано уравнение
$$x^{8} = -3$$
Т.к. степень в ур-нии равна = 8 и свободный член = -3 < 0,
зн. действительных решений у соотв. ур-ния не существует

Остальные 8 корня(ей) являются комплексными.
сделаем замену:
$$z = x$$
тогда ур-ние будет таким:
$$z^{8} = -3$$
Любое комплексное число можно представить так:
$$z = r e^{i p}$$
подставляем в уравнение
$$r^{8} e^{8 i p} = -3$$
где
$$r = \sqrt[8]{3}$$
- модуль комплексного числа
Подставляем r:
$$e^{8 i p} = -1$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = -1$$
значит
$$\cos{\left(8 p \right)} = -1$$
и
$$\sin{\left(8 p \right)} = 0$$
тогда
$$p = \frac{\pi N}{4} + \frac{\pi}{8}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для z
Значит, решением будет для z:
$$z_{1} = - \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{2} = \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{3} = - \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{4} = \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{5} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{6} = \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{7} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{8} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
делаем обратную замену
$$z = x$$
$$x = z$$

Тогда, окончательный ответ:
$$x_{1} = - \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{2} = \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{3} = - \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{4} = \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{5} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{6} = \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$x_{7} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{8} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$

Быстрый ответ

                  ___________                ___________
                 /       ___                /       ___ 
       8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  
x1 = - \/ 3 *  /   - - -----  + I*\/ 3 *  /   - + ----- 
             \/    2     4              \/    2     4   

$$x_{1} = - \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$

                ___________                ___________
               /       ___                /       ___ 
     8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  
x2 = \/ 3 *  /   - - -----  - I*\/ 3 *  /   - + ----- 
           \/    2     4              \/    2     4   

$$x_{2} = \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$

                  ___________                ___________
                 /       ___                /       ___ 
       8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  
x3 = - \/ 3 *  /   - + -----  - I*\/ 3 *  /   - - ----- 
             \/    2     4              \/    2     4   

$$x_{3} = - \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$

                ___________                ___________
               /       ___                /       ___ 
     8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  
x4 = \/ 3 *  /   - + -----  + I*\/ 3 *  /   - - ----- 
           \/    2     4              \/    2     4   

$$x_{4} = \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$

       /                 ___________                    ___________\                    ___________                    ___________
       |                /       ___                    /       ___ |                   /       ___                    /       ___ 
       |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  
       |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- 
       |            \/    2     4                  \/    2     4   |               \/    2     4                  \/    2     4   
x5 = I*|---------------------------- + ----------------------------| + ---------------------------- - ----------------------------
       \             2                              2              /                2                              2              

$$x_{5} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

       /                 ___________                    ___________\                    ___________                    ___________
       |                /       ___                    /       ___ |                   /       ___                    /       ___ 
       |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  
       |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- 
       |            \/    2     4                  \/    2     4   |               \/    2     4                  \/    2     4   
x6 = I*|---------------------------- - ----------------------------| + ---------------------------- + ----------------------------
       \             2                              2              /                2                              2              

$$x_{6} = \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$

       /                 ___________                    ___________\                    ___________                    ___________
       |                /       ___                    /       ___ |                   /       ___                    /       ___ 
       |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  
       |\/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- 
       |            \/    2     4                  \/    2     4   |               \/    2     4                  \/    2     4   
x7 = I*|---------------------------- - ----------------------------| - ---------------------------- - ----------------------------
       \             2                              2              /                2                              2              

$$x_{7} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

       /                   ___________                    ___________\                    ___________                    ___________
       |                  /       ___                    /       ___ |                   /       ___                    /       ___ 
       |    ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  
       |  \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- 
       |              \/    2     4                  \/    2     4   |               \/    2     4                  \/    2     4   
x8 = I*|- ---------------------------- - ----------------------------| + ---------------------------- - ----------------------------
       \               2                              2              /                2                              2              

$$x_{8} = - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$

Сумма и произведение корней

сумма

                                                                                                                                                                                                                          /                 ___________                    ___________\                    ___________                    ___________     /                 ___________                    ___________\                    ___________                    ___________     /                 ___________                    ___________\                    ___________                    ___________     /                   ___________                    ___________\                    ___________                    ___________
                                                                                                                                                                                                                          |                /       ___                    /       ___ |                   /       ___                    /       ___      |                /       ___                    /       ___ |                   /       ___                    /       ___      |                /       ___                    /       ___ |                   /       ___                    /       ___      |                  /       ___                    /       ___ |                   /       ___                    /       ___ 
                 ___________                ___________              ___________                ___________                ___________                ___________              ___________                ___________     |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2       |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2       |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2       |    ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  
                /       ___                /       ___              /       ___                /       ___                /       ___                /       ___              /       ___                /       ___      |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 3 *  /   - - -----      |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----      |\/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + -----      |  \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- 
      8 ___    /  1   \/ 2       8 ___    /  1   \/ 2     8 ___    /  1   \/ 2       8 ___    /  1   \/ 2       8 ___    /  1   \/ 2       8 ___    /  1   \/ 2     8 ___    /  1   \/ 2       8 ___    /  1   \/ 2       |            \/    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   
0 + - \/ 3 *  /   - - -----  + I*\/ 3 *  /   - + -----  + \/ 3 *  /   - - -----  - I*\/ 3 *  /   - + -----  + - \/ 3 *  /   - + -----  - I*\/ 3 *  /   - - -----  + \/ 3 *  /   - + -----  + I*\/ 3 *  /   - - -----  + I*|---------------------------- + ----------------------------| + ---------------------------- - ---------------------------- + I*|---------------------------- - ----------------------------| + ---------------------------- + ---------------------------- + I*|---------------------------- - ----------------------------| - ---------------------------- - ---------------------------- + I*|- ---------------------------- - ----------------------------| + ---------------------------- - ----------------------------
            \/    2     4              \/    2     4            \/    2     4              \/    2     4              \/    2     4              \/    2     4            \/    2     4              \/    2     4        \             2                              2              /                2                              2                   \             2                              2              /                2                              2                   \             2                              2              /                2                              2                   \               2                              2              /                2                              2              

$$\left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) - \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - i \left(\frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) - i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) - i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)$$

=

  /                 ___________                    ___________\     /                 ___________                    ___________\     /                 ___________                    ___________\     /                   ___________                    ___________\
  |                /       ___                    /       ___ |     |                /       ___                    /       ___ |     |                /       ___                    /       ___ |     |                  /       ___                    /       ___ |
  |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     |    ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |
  |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |     |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |     |\/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- |     |  \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 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              /

$$i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) + i \left(\frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

произведение

                                                                                                                                                                                                                      /  /                 ___________                    ___________\                    ___________                    ___________\ /  /                 ___________                    ___________\                    ___________                    ___________\ /  /                 ___________                    ___________\                    ___________                    ___________\ /  /                   ___________                    ___________\                    ___________                    ___________\
                                                                                                                                                                                                                      |  |                /       ___                    /       ___ |                   /       ___                    /       ___ | |  |                /       ___                    /       ___ |                   /       ___                    /       ___ | |  |                /       ___                    /       ___ |                   /       ___                    /       ___ | |  |                  /       ___                    /       ___ |                   /       ___                    /       ___ |
  /             ___________                ___________\ /           ___________                ___________\ /             ___________                ___________\ /           ___________                ___________\ |  |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  | |  |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  | |  |  ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  | |  |    ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |     ___ 8 ___    /  1   \/ 2       ___ 8 ___    /  1   \/ 2  |
  |            /       ___                /       ___ | |          /       ___                /       ___ | |            /       ___                /       ___ | |          /       ___                /       ___ | |  |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- | |  |\/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- | |  |\/ 2 *\/ 3 *  /   - + -----    \/ 2 *\/ 3 *  /   - - ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- | |  |  \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |   \/ 2 *\/ 3 *  /   - - -----    \/ 2 *\/ 3 *  /   - + ----- |
  |  8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  | |8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  | |  8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  | |8 ___    /  1   \/ 2       8 ___    /  1   \/ 2  | |  |            \/    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   |
1*|- \/ 3 *  /   - - -----  + I*\/ 3 *  /   - + ----- |*|\/ 3 *  /   - - -----  - I*\/ 3 *  /   - + ----- |*|- \/ 3 *  /   - + -----  - I*\/ 3 *  /   - - ----- |*|\/ 3 *  /   - + -----  + I*\/ 3 *  /   - - ----- |*|I*|---------------------------- + ----------------------------| + ---------------------------- - ----------------------------|*|I*|---------------------------- - ----------------------------| + ---------------------------- + ----------------------------|*|I*|---------------------------- - ----------------------------| - ---------------------------- - ----------------------------|*|I*|- ---------------------------- - ----------------------------| + ---------------------------- - ----------------------------|
  \        \/    2     4              \/    2     4   / \      \/    2     4              \/    2     4   / \        \/    2     4              \/    2     4   / \      \/    2     4              \/    2     4   / \  \             2                              2              /                2                              2              / \  \             2                              2              /                2                              2              / \  \             2                              2              /                2                              2              / \  \               2                              2              /                2                              2              /

$$1 \left(- \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{3} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{3} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(\frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)$$

=

3

$$3$$

Численный ответ

x1 = 0.439015463195998 - 1.05987708533928*i

x2 = 1.05987708533928 - 0.439015463195998*i

x3 = -0.439015463195998 - 1.05987708533928*i

x4 = 0.439015463195998 + 1.05987708533928*i

x5 = -1.05987708533928 - 0.439015463195998*i

x6 = -1.05987708533928 + 0.439015463195998*i

x7 = 1.05987708533928 + 0.439015463195998*i

x8 = -0.439015463195998 + 1.05987708533928*i

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