z^5=1+i (уравнение)

Найду корень уравнения: z^5=1+i

Вы ввели

 5        
z  = 1 + I

$$z^{5} = 1 + i$$

Подробное решение

Дано уравнение
$$z^{5} = 1 + i$$
Т.к. степень в ур-нии равна = 5 - не содержит чётного числа в числителе, то
ур-ние будет иметь один действительный корень.
Извлечём корень 5-й степени из обеих частей ур-ния:
Получим:
$$\sqrt[5]{z^{5}} = \sqrt[5]{1 + i}$$
или
$$z = \sqrt[5]{1 + i}$$
Раскрываем скобочки в правой части ур-ния

z = 1+i^1/5

Получим ответ: z = (1 + i)^(1/5)

Остальные 5 корня(ей) являются комплексными.
сделаем замену:
$$w = z$$
тогда ур-ние будет таким:
$$w^{5} = 1 + i$$
Любое комплексное число можно представить так:
$$w = r e^{i p}$$
подставляем в уравнение
$$r^{5} e^{5 i p} = 1 + i$$
где
$$r = \sqrt[10]{2}$$
- модуль комплексного числа
Подставляем r:
$$e^{5 i p} = \frac{\sqrt{2} \left(1 + i\right)}{2}$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = \frac{\sqrt{2} \left(1 + i\right)}{2}$$
значит
$$\cos{\left(5 p \right)} = \frac{\sqrt{2}}{2}$$
и
$$\sin{\left(5 p \right)} = \frac{\sqrt{2}}{2}$$
тогда
$$p = \frac{2 \pi N}{5} + \frac{\pi}{20}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для w
Значит, решением будет для w:
$$w_{1} = - \frac{2^{\frac{3}{5}}}{2} - \frac{2^{\frac{3}{5}} i}{2}$$
$$w_{2} = \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{3}{5}} i}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} i}{8}$$
$$w_{3} = - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{2^{\frac{3}{5}} \sqrt{5}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} i}{8} + \frac{2^{\frac{3}{5}} i}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}$$
$$w_{4} = - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} i}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
$$w_{5} = - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{2^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} i}{8}$$
делаем обратную замену
$$w = z$$
$$z = w$$

Тогда, окончательный ответ:
$$z_{1} = - \frac{2^{\frac{3}{5}}}{2} - \frac{2^{\frac{3}{5}} i}{2}$$
$$z_{2} = \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{3}{5}} i}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} i}{8}$$
$$z_{3} = - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{2^{\frac{3}{5}} \sqrt{5}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} i}{8} + \frac{2^{\frac{3}{5}} i}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}$$
$$z_{4} = - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} i}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
$$z_{5} = - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{2^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} i}{8}$$

Быстрый ответ

        3/5      3/5
       2      I*2   
z1 = - ---- - ------
        2       2   

$$z_{1} = - \frac{2^{\frac{3}{5}}}{2} - \frac{2^{\frac{3}{5}} i}{2}$$

              /                 ___________             \             ___________             
              |                /       ___              |            /       ___              
              |        3/5    /  5   \/ 5               |    3/5    /  5   \/ 5               
      3/5     | 3/5   2   *  /   - - -----     3/5   ___|   2   *  /   - - -----     3/5   ___
     2        |2           \/    8     8      2   *\/ 5 |        \/    8     8      2   *\/ 5 
z2 = ---- + I*|---- - --------------------- + ----------| + --------------------- + ----------
      8       \ 8               2                 8     /             2                 8     

$$z_{2} = \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8}\right)$$

              /                              ___________                   ___________\             ___________                                ___________
              |                             /       ___                   /       ___ |            /       ___                                /       ___ 
              |                     3/5    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5                   3/5   ___    /  5   \/ 5  
      3/5     | 3/5    3/5   ___   2   *  /   - - -----    2   *\/ 5 *  /   - - ----- |   2   *  /   - - -----     3/5   ___   2   *\/ 5 *  /   - - ----- 
     2        |2      2   *\/ 5         \/    8     8                 \/    8     8   |        \/    8     8      2   *\/ 5               \/    8     8   
z3 = ---- + I*|---- - ---------- + --------------------- + ---------------------------| - --------------------- - ---------- - ---------------------------
      8       \ 8         8                  4                          4             /             4                 8                     4             

$$z_{3} = - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{2^{\frac{3}{5}} \sqrt{5}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}}}{8} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right)$$

              /                 ___________             ___________             ___________      ___________                   ___________                   ___________\             ___________             ___________             ___________      ___________                   ___________                   ___________
              |                /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ |            /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ 
              |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  
      3/5     | 3/5   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - + -----    2   *\/ 5 *  /   - - ----- 
     2        |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |        \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   
z4 = ---- + I*|---- + --------------------- + --------------------- + -------------------------------------- - --------------------------- + ---------------------------| - --------------------- - --------------------- + -------------------------------------- - --------------------------- + ---------------------------
      8       \ 8               8                       8                               2                                   8                             8             /             8                       8                               2                                   8                             8             

$$z_{4} = - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)$$

              /                 ___________             ___________             ___________      ___________                   ___________                   ___________\             ___________             ___________             ___________      ___________                   ___________                   ___________
              |                /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ |            /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ 
              |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  
      3/5     | 3/5   2   *  /   - + -----    2   *  /   - - -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- 
     2        |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |        \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   
z5 = ---- + I*|---- - --------------------- + --------------------- - -------------------------------------- - --------------------------- - ---------------------------| - --------------------- + --------------------- - -------------------------------------- + --------------------------- + ---------------------------
      8       \ 8               8                       8                               2                                   8                             8             /             8                       8                               2                                   8                             8             

$$z_{5} = - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8}\right)$$

Сумма и произведение корней

сумма

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                           |        3/5    /  5   \/ 5               |    3/5    /  5   \/ 5                           |                     3/5    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5                   3/5   ___    /  5   \/ 5              |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5              |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  
   3/5      3/5    3/5     | 3/5   2   *  /   - - -----     3/5   ___|   2   *  /   - - -----     3/5   ___    3/5     | 3/5    3/5   ___   2   *  /   - - -----    2   *\/ 5 *  /   - - ----- |   2   *  /   - - -----     3/5   ___   2   *\/ 5 *  /   - - -----     3/5     | 3/5   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - + -----    2   *\/ 5 *  /   - - -----     3/5     | 3/5   2   *  /   - + -----    2   *  /   - - -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- 
  2      I*2      2        |2           \/    8     8      2   *\/ 5 |        \/    8     8      2   *\/ 5    2        |2      2   *\/ 5         \/    8     8                 \/    8     8   |        \/    8     8      2   *\/ 5               \/    8     8      2        |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |        \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8      2        |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |        \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   
- ---- - ------ + ---- + I*|---- - --------------------- + ----------| + --------------------- + ---------- + ---- + I*|---- - ---------- + --------------------- + ---------------------------| - --------------------- - ---------- - --------------------------- + ---- + I*|---- + --------------------- + --------------------- + -------------------------------------- - --------------------------- + ---------------------------| - --------------------- - --------------------- + -------------------------------------- - --------------------------- + --------------------------- + ---- + I*|---- - --------------------- + --------------------- - -------------------------------------- - --------------------------- - ---------------------------| - --------------------- + --------------------- - -------------------------------------- + --------------------------- + ---------------------------
   2       2       8       \ 8               2                 8     /             2                 8         8       \ 8         8                  4                          4             /             4                 8                     4                 8       \ 8               8                       8                               2                                   8                             8             /             8                       8                               2                                   8                             8                 8       \ 8               8                       8                               2                                   8                             8             /             8                       8                               2                                   8                             8             

$$\left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8}\right)\right) + \left(\left(\left(\left(- \frac{2^{\frac{3}{5}}}{2} - \frac{2^{\frac{3}{5}} i}{2}\right) + \left(\frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8}\right)\right)\right) + \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{2^{\frac{3}{5}} \sqrt{5}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}}}{8} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right)\right)\right) + \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right)\right)$$

=

  /                 ___________             \     /                              ___________                   ___________\     /                 ___________             ___________             ___________      ___________                   ___________                   ___________\     /                 ___________             ___________             ___________      ___________                   ___________                   ___________\         
  |                /       ___              |     |                             /       ___                   /       ___ |     |                /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ |     |                /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ |         
  |        3/5    /  5   \/ 5               |     |                     3/5    /  5   \/ 5      3/5   ___    /  5   \/ 5  |     |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |     |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |         
  | 3/5   2   *  /   - - -----     3/5   ___|     | 3/5    3/5   ___   2   *  /   - - -----    2   *\/ 5 *  /   - - ----- |     | 3/5   2   *  /   - + -----    2   *  /   - - -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |     | 3/5   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |      3/5
  |2           \/    8     8      2   *\/ 5 |     |2      2   *\/ 5         \/    8     8                 \/    8     8   |     |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |     |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |   I*2   
I*|---- - --------------------- + ----------| + I*|---- - ---------- + --------------------- + ---------------------------| + I*|---- - --------------------- + --------------------- - -------------------------------------- - --------------------------- - ---------------------------| + I*|---- + --------------------- + --------------------- + -------------------------------------- - --------------------------- + ---------------------------| - ------
  \ 8               2                 8     /     \ 8         8                  4                          4             /     \ 8               8                       8                               2                                   8                             8             /     \ 8               8                       8                               2                                   8                             8             /     2   

$$i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8}\right) - \frac{2^{\frac{3}{5}} i}{2} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8}\right) + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right) + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)$$

произведение

                  /         /                 ___________             \             ___________             \ /         /                              ___________                   ___________\             ___________                                ___________\ /         /                 ___________             ___________             ___________      ___________                   ___________                   ___________\             ___________             ___________             ___________      ___________                   ___________                   ___________\ /         /                 ___________             ___________             ___________      ___________                   ___________                   ___________\             ___________             ___________             ___________      ___________                   ___________                   ___________\
                  |         |                /       ___              |            /       ___              | |         |                             /       ___                   /       ___ |            /       ___                                /       ___ | |         |                /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ |            /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ | |         |                /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ |            /       ___             /       ___             /       ___      /       ___                   /       ___                   /       ___ |
                  |         |        3/5    /  5   \/ 5               |    3/5    /  5   \/ 5               | |         |                     3/5    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5                   3/5   ___    /  5   \/ 5  | |         |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  | |         |        3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |    3/5    /  5   \/ 5      3/5    /  5   \/ 5      3/5    /  5   \/ 5      /  5   \/ 5      3/5   ___    /  5   \/ 5      3/5   ___    /  5   \/ 5  |
/   3/5      3/5\ | 3/5     | 3/5   2   *  /   - - -----     3/5   ___|   2   *  /   - - -----     3/5   ___| | 3/5     | 3/5    3/5   ___   2   *  /   - - -----    2   *\/ 5 *  /   - - ----- |   2   *  /   - - -----     3/5   ___   2   *\/ 5 *  /   - - ----- | | 3/5     | 3/5   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - + -----    2   *\/ 5 *  /   - - ----- | | 3/5     | 3/5   2   *  /   - + -----    2   *  /   - - -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |   2   *  /   - - -----    2   *  /   - + -----    2   *  /   - - ----- *  /   - + -----    2   *\/ 5 *  /   - - -----    2   *\/ 5 *  /   - + ----- |
|  2      I*2   | |2        |2           \/    8     8      2   *\/ 5 |        \/    8     8      2   *\/ 5 | |2        |2      2   *\/ 5         \/    8     8                 \/    8     8   |        \/    8     8      2   *\/ 5               \/    8     8   | |2        |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |        \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   | |2        |2           \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |        \/    8     8           \/    8     8           \/    8     8    \/    8     8                 \/    8     8                 \/    8     8   |
|- ---- - ------|*|---- + I*|---- - --------------------- + ----------| + --------------------- + ----------|*|---- + I*|---- - ---------- + --------------------- + ---------------------------| - --------------------- - ---------- - ---------------------------|*|---- + I*|---- + --------------------- + --------------------- + -------------------------------------- - --------------------------- + ---------------------------| - --------------------- - --------------------- + -------------------------------------- - --------------------------- + ---------------------------|*|---- + I*|---- - --------------------- + --------------------- - -------------------------------------- - --------------------------- - ---------------------------| - --------------------- + --------------------- - -------------------------------------- + --------------------------- + ---------------------------|
\   2       2   / \ 8       \ 8               2                 8     /             2                 8     / \ 8       \ 8         8                  4                          4             /             4                 8                     4             / \ 8       \ 8               8                       8                               2                                   8                             8             /             8                       8                               2                                   8                             8             / \ 8       \ 8               8                       8                               2                                   8                             8             /             8                       8                               2                                   8                             8             /

$$\left(- \frac{2^{\frac{3}{5}}}{2} - \frac{2^{\frac{3}{5}} i}{2}\right) \left(\frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5}}{8}\right)\right) \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} - \frac{2^{\frac{3}{5}} \sqrt{5}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}}}{8} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right)\right) \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right) \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + i \left(- \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} - \frac{2^{\frac{3}{5}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{8} + \frac{2^{\frac{3}{5}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{8} + \frac{2^{\frac{3}{5}}}{8}\right)\right)$$

=

             ______________      ________________      _______________      _______________      ________________        _______________
            /          ___      /            ___      /           ___      /           ___      /            ___        /           ___ 
        5*\/  10 - 2*\/ 5     \/  -50 - 10*\/ 5     \/  50 - 10*\/ 5     \/  50 + 10*\/ 5     \/  -50 + 10*\/ 5     5*\/  -10 + 2*\/ 5  
1 + I - ------------------- - ------------------- - ------------------ + ------------------ + ------------------- + --------------------
                 64                    32                   64                   32                    64                    64         

$$- \frac{5 \sqrt{10 - 2 \sqrt{5}}}{64} - \frac{\sqrt{50 - 10 \sqrt{5}}}{64} + \frac{\sqrt{10 \sqrt{5} + 50}}{32} + 1 - \frac{\sqrt{-50 - 10 \sqrt{5}}}{32} + \frac{\sqrt{-50 + 10 \sqrt{5}}}{64} + \frac{5 \sqrt{-10 + 2 \sqrt{5}}}{64} + i$$

Численный ответ

z1 = -0.757858283255199 - 0.757858283255199*i

z2 = 0.167662308256181 + 1.05857815270638*i

z3 = -0.954957147571795 + 0.486574969864436*i

z4 = 1.05857815270638 + 0.167662308256181*i

z5 = 0.486574969864436 - 0.954957147571795*i