z^8=-1 (уравнение)

Найду корень уравнения: z^8=-1

Вы ввели

 8     
z  = -1

$$z^{8} = -1$$

Подробное решение

Дано уравнение
$$z^{8} = -1$$
Т.к. степень в ур-нии равна = 8 и свободный член = -1 < 0,
зн. действительных решений у соотв. ур-ния не существует

Остальные 8 корня(ей) являются комплексными.
сделаем замену:
$$w = z$$
тогда ур-ние будет таким:
$$w^{8} = -1$$
Любое комплексное число можно представить так:
$$w = r e^{i p}$$
подставляем в уравнение
$$r^{8} e^{8 i p} = -1$$
где
$$r = 1$$
- 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 w
Значит, решением будет для w:
$$w_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$w_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$w_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$w_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$w_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$w_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
$$w_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}$$
$$w_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}$$
делаем обратную замену
$$w = z$$
$$z = w$$

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

Быстрый ответ

            ___________          ___________
           /       ___          /       ___ 
          /  1   \/ 2          /  1   \/ 2  
z1 = -   /   - - -----  + I*  /   - + ----- 
       \/    2     4        \/    2     4   

$$z_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$

          ___________          ___________
         /       ___          /       ___ 
        /  1   \/ 2          /  1   \/ 2  
z2 =   /   - - -----  - I*  /   - + ----- 
     \/    2     4        \/    2     4   

$$z_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$

            ___________          ___________
           /       ___          /       ___ 
          /  1   \/ 2          /  1   \/ 2  
z3 = -   /   - + -----  - I*  /   - - ----- 
       \/    2     4        \/    2     4   

$$z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$

          ___________          ___________
         /       ___          /       ___ 
        /  1   \/ 2          /  1   \/ 2  
z4 =   /   - + -----  + I*  /   - - ----- 
     \/    2     4        \/    2     4   

$$z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$

       /           ___________              ___________\              ___________              ___________
       |          /       ___              /       ___ |             /       ___              /       ___ 
       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - + -----    \/ 2 *  /   - - ----- 
       |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
z5 = I*|---------------------- + ----------------------| + ---------------------- - ----------------------
       \          2                        2           /             2                        2           

$$z_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

       /           ___________              ___________\              ___________              ___________
       |          /       ___              /       ___ |             /       ___              /       ___ 
       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
       |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
z6 = I*|---------------------- - ----------------------| + ---------------------- + ----------------------
       \          2                        2           /             2                        2           

$$z_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$

       /           ___________              ___________\              ___________              ___________
       |          /       ___              /       ___ |             /       ___              /       ___ 
       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |\/ 2 *  /   - + -----    \/ 2 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
       |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
z7 = I*|---------------------- - ----------------------| - ---------------------- - ----------------------
       \          2                        2           /             2                        2           

$$z_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

       /             ___________              ___________\              ___________              ___________
       |            /       ___              /       ___ |             /       ___              /       ___ 
       |    ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |  \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
       |        \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
z8 = I*|- ---------------------- - ----------------------| + ---------------------- - ----------------------
       \            2                        2           /             2                        2           

$$z_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)$$

Сумма и произведение корней

сумма

                                                                                                                                                                      /           ___________              ___________\              ___________              ___________     /           ___________              ___________\              ___________              ___________     /           ___________              ___________\              ___________              ___________     /             ___________              ___________\              ___________              ___________
                                                                                                                                                                      |          /       ___              /       ___ |             /       ___              /       ___      |          /       ___              /       ___ |             /       ___              /       ___      |          /       ___              /       ___ |             /       ___              /       ___      |            /       ___              /       ___ |             /       ___              /       ___ 
       ___________          ___________        ___________          ___________          ___________          ___________        ___________          ___________     |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2       |    ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
      /       ___          /       ___        /       ___          /       ___          /       ___          /       ___        /       ___          /       ___      |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - + -----    \/ 2 *  /   - - -----      |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + -----      |\/ 2 *  /   - + -----    \/ 2 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + -----      |  \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
     /  1   \/ 2          /  1   \/ 2        /  1   \/ 2          /  1   \/ 2          /  1   \/ 2          /  1   \/ 2        /  1   \/ 2          /  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   
-   /   - - -----  + I*  /   - + -----  +   /   - - -----  - I*  /   - + -----  + -   /   - + -----  - I*  /   - - -----  +   /   - + -----  + I*  /   - - -----  + 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           

$$\left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) + \left(\left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(\left(\left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right)\right) + \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) + \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)\right)\right)\right)$$

=

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

$$i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$

произведение

                                                                                                                                                                    /  /           ___________              ___________\              ___________              ___________\ /  /           ___________              ___________\              ___________              ___________\ /  /           ___________              ___________\              ___________              ___________\ /  /             ___________              ___________\              ___________              ___________\
                                                                                                                                                                    |  |          /       ___              /       ___ |             /       ___              /       ___ | |  |          /       ___              /       ___ |             /       ___              /       ___ | |  |          /       ___              /       ___ |             /       ___              /       ___ | |  |            /       ___              /       ___ |             /       ___              /       ___ |
/       ___________          ___________\ /     ___________          ___________\ /       ___________          ___________\ /     ___________          ___________\ |  |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  | |  |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  | |  |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  | |  |    ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  |
|      /       ___          /       ___ | |    /       ___          /       ___ | |      /       ___          /       ___ | |    /       ___          /       ___ | |  |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - + -----    \/ 2 *  /   - - ----- | |  |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- | |  |\/ 2 *  /   - + -----    \/ 2 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- | |  |  \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |
|     /  1   \/ 2          /  1   \/ 2  | |   /  1   \/ 2          /  1   \/ 2  | |     /  1   \/ 2          /  1   \/ 2  | |   /  1   \/ 2          /  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   |
|-   /   - - -----  + I*  /   - + ----- |*|  /   - - -----  - I*  /   - + ----- |*|-   /   - + -----  - I*  /   - - ----- |*|  /   - + -----  + I*  /   - - ----- |*|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           /

$$\left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)$$

=

1

$$1$$

Численный ответ

z1 = 0.923879532511287 - 0.38268343236509*i

z2 = -0.38268343236509 - 0.923879532511287*i

z3 = -0.923879532511287 + 0.38268343236509*i

z4 = 0.38268343236509 - 0.923879532511287*i

z5 = -0.38268343236509 + 0.923879532511287*i

z6 = -0.923879532511287 - 0.38268343236509*i

z7 = 0.923879532511287 + 0.38268343236509*i

z8 = 0.38268343236509 + 0.923879532511287*i

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