x^8+24=0 (уравнение)

Найду корень уравнения: x^8+24=0

Вы ввели

 8         
x  + 24 = 0

$$x^{8} + 24 = 0$$

Подробное решение

Дано уравнение
$$x^{8} + 24 = 0$$
Т.к. степень в ур-нии равна = 8 и свободный член = -24 < 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:
$$z^{8} = -24$$
Любое комплексное число можно представить так:
$$z = r e^{i p}$$
подставляем в уравнение
$$r^{8} e^{8 i p} = -24$$
где
$$r = \sqrt[8]{24}$$
- 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:
$$z_{1} = - \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{2} = \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{3} = - \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{4} = \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{5} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{6} = \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$z_{7} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$z_{8} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
делаем обратную замену
$$z = x$$
$$x = z$$

Тогда, окончательный ответ:
$$x_{1} = - \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{2} = \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$x_{3} = - \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{4} = \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$x_{5} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{6} = \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$x_{7} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$x_{8} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$

Быстрый ответ

                   ___________                 ___________
                  /       ___                 /       ___ 
       8 ____    /  1   \/ 2       8 ____    /  1   \/ 2  
x1 = - \/ 24 *  /   - - -----  + I*\/ 24 *  /   - + ----- 
              \/    2     4               \/    2     4   

$$x_{1} = - \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$

                 ___________                 ___________
                /       ___                 /       ___ 
     8 ____    /  1   \/ 2       8 ____    /  1   \/ 2  
x2 = \/ 24 *  /   - - -----  - I*\/ 24 *  /   - + ----- 
            \/    2     4               \/    2     4   

$$x_{2} = \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$

                   ___________                 ___________
                  /       ___                 /       ___ 
       8 ____    /  1   \/ 2       8 ____    /  1   \/ 2  
x3 = - \/ 24 *  /   - + -----  - I*\/ 24 *  /   - - ----- 
              \/    2     4               \/    2     4   

$$x_{3} = - \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$

                 ___________                 ___________
                /       ___                 /       ___ 
     8 ____    /  1   \/ 2       8 ____    /  1   \/ 2  
x4 = \/ 24 *  /   - + -----  + I*\/ 24 *  /   - - ----- 
            \/    2     4               \/    2     4   

$$x_{4} = \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$

       /                ___________                   ___________\                   ___________                   ___________
       |               /       ___                   /       ___ |                  /       ___                   /       ___ 
       | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  
       |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - + -----    2   *\/ 6 *  /   - - ----- 
       |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   
x5 = I*|--------------------------- + ---------------------------| + --------------------------- - ---------------------------
       \             2                             2             /                2                             2             

$$x_{5} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

       /                ___________                   ___________\                   ___________                   ___________
       |               /       ___                   /       ___ |                  /       ___                   /       ___ 
       | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  
       |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + ----- 
       |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   
x6 = I*|--------------------------- - ---------------------------| + --------------------------- + ---------------------------
       \             2                             2             /                2                             2             

$$x_{6} = \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$

       /                ___________                   ___________\                   ___________                   ___________
       |               /       ___                   /       ___ |                  /       ___                   /       ___ 
       | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  
       |2   *\/ 3 *  /   - + -----    2   *\/ 3 *  /   - - ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + ----- 
       |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   
x7 = I*|--------------------------- - ---------------------------| - --------------------------- - ---------------------------
       \             2                             2             /                2                             2             

$$x_{7} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

       /                  ___________                   ___________\                   ___________                   ___________
       |                 /       ___                   /       ___ |                  /       ___                   /       ___ 
       |   7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  
       |  2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + ----- 
       |             \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   
x8 = I*|- --------------------------- - ---------------------------| + --------------------------- - ---------------------------
       \               2                             2             /                2                             2             

$$x_{8} = - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$

Сумма и произведение корней

сумма

                                                                                                                                                                                                                              /                ___________                   ___________\                   ___________                   ___________     /                ___________                   ___________\                   ___________                   ___________     /                ___________                   ___________\                   ___________                   ___________     /                  ___________                   ___________\                   ___________                   ___________
                                                                                                                                                                                                                              |               /       ___                   /       ___ |                  /       ___                   /       ___      |               /       ___                   /       ___ |                  /       ___                   /       ___      |               /       ___                   /       ___ |                  /       ___                   /       ___      |                 /       ___                   /       ___ |                  /       ___                   /       ___ 
              ___________                 ___________               ___________                 ___________                 ___________                 ___________               ___________                 ___________     | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2       | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2       | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2       |   7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  
             /       ___                 /       ___               /       ___                 /       ___                 /       ___                 /       ___               /       ___                 /       ___      |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - + -----    2   *\/ 6 *  /   - - -----      |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + -----      |2   *\/ 3 *  /   - + -----    2   *\/ 3 *  /   - - ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + -----      |  2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + ----- 
  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   
- \/ 24 *  /   - - -----  + I*\/ 24 *  /   - + -----  + \/ 24 *  /   - - -----  - I*\/ 24 *  /   - + -----  + - \/ 24 *  /   - + -----  - I*\/ 24 *  /   - - -----  + \/ 24 *  /   - + -----  + I*\/ 24 *  /   - - -----  + 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             

$$\left(- \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(- \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) + \left(\left(\frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(\left(\left(- \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(\sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(- \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right)\right) + \left(\sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) + \left(- \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)\right)\right)\right)$$

=

  /                ___________                   ___________\     /                ___________                   ___________\     /                ___________                   ___________\     /                  ___________                   ___________\
  |               /       ___                   /       ___ |     |               /       ___                   /       ___ |     |               /       ___                   /       ___ |     |                 /       ___                   /       ___ |
  | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |     | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |     | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |     |   7/8 8 ___    /  1   \/ 2      7/8 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             /

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

произведение

                                                                                                                                                                                                                            /  /                ___________                   ___________\                   ___________                   ___________\ /  /                ___________                   ___________\                   ___________                   ___________\ /  /                ___________                   ___________\                   ___________                   ___________\ /  /                  ___________                   ___________\                   ___________                   ___________\
                                                                                                                                                                                                                            |  |               /       ___                   /       ___ |                  /       ___                   /       ___ | |  |               /       ___                   /       ___ |                  /       ___                   /       ___ | |  |               /       ___                   /       ___ |                  /       ___                   /       ___ | |  |                 /       ___                   /       ___ |                  /       ___                   /       ___ |
/              ___________                 ___________\ /            ___________                 ___________\ /              ___________                 ___________\ /            ___________                 ___________\ |  | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  | |  | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  | |  | 7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  | |  |   7/8 8 ___    /  1   \/ 2      7/8 8 ___    /  1   \/ 2  |    3/4 8 ___    /  1   \/ 2      3/4 8 ___    /  1   \/ 2  |
|             /       ___                 /       ___ | |           /       ___                 /       ___ | |             /       ___                 /       ___ | |           /       ___                 /       ___ | |  |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - + -----    2   *\/ 6 *  /   - - ----- | |  |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + ----- | |  |2   *\/ 3 *  /   - + -----    2   *\/ 3 *  /   - - ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + ----- | |  |  2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 6 *  /   - - -----    2   *\/ 6 *  /   - + ----- |
|  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   |
|- \/ 24 *  /   - - -----  + I*\/ 24 *  /   - + ----- |*|\/ 24 *  /   - - -----  - I*\/ 24 *  /   - + ----- |*|- \/ 24 *  /   - + -----  - I*\/ 24 *  /   - - ----- |*|\/ 24 *  /   - + -----  + I*\/ 24 *  /   - - ----- |*|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             /

$$\left(- \sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt[8]{24} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[8]{24} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt[8]{24} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[8]{24} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(\frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[8]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{7}{8}} \sqrt[8]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)$$

=

24

$$24$$

Численный ответ

x1 = 0.569332617776004 + 1.37449052733621*i

x2 = 1.37449052733621 - 0.569332617776004*i

x3 = -1.37449052733621 - 0.569332617776004*i

x4 = -1.37449052733621 + 0.569332617776004*i

x5 = -0.569332617776004 - 1.37449052733621*i

x6 = 1.37449052733621 + 0.569332617776004*i

x7 = 0.569332617776004 - 1.37449052733621*i

x8 = -0.569332617776004 + 1.37449052733621*i