z^12=-4 (уравнение)

Найду корень уравнения: z^12=-4

Вы ввели

 12     
z   = -4

$$z^{12} = -4$$

Подробное решение

Дано уравнение
$$z^{12} = -4$$
Т.к. степень в ур-нии равна = 12 и свободный член = -4 < 0,
зн. действительных решений у соотв. ур-ния не существует

Остальные 12 корня(ей) являются комплексными.
сделаем замену:
$$w = z$$
тогда ур-ние будет таким:
$$w^{12} = -4$$
Любое комплексное число можно представить так:
$$w = r e^{i p}$$
подставляем в уравнение
$$r^{12} e^{12 i p} = -4$$
где
$$r = \sqrt[6]{2}$$
- модуль комплексного числа
Подставляем r:
$$e^{12 i p} = -1$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(12 p \right)} + \cos{\left(12 p \right)} = -1$$
значит
$$\cos{\left(12 p \right)} = -1$$
и
$$\sin{\left(12 p \right)} = 0$$
тогда
$$p = \frac{\pi N}{6} + \frac{\pi}{12}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для w
Значит, решением будет для w:
$$w_{1} = - \frac{2^{\frac{2}{3}}}{2} - \frac{2^{\frac{2}{3}} i}{2}$$
$$w_{2} = - \frac{2^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} i}{2}$$
$$w_{3} = \frac{2^{\frac{2}{3}}}{2} - \frac{2^{\frac{2}{3}} i}{2}$$
$$w_{4} = \frac{2^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} i}{2}$$
$$w_{5} = - \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$w_{6} = - \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} - \frac{2^{\frac{2}{3}} i}{4}$$
$$w_{7} = \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$w_{8} = \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} + \frac{2^{\frac{2}{3}} i}{4}$$
$$w_{9} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4} - \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$w_{10} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} + \frac{2^{\frac{2}{3}} i}{4}$$
$$w_{11} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$w_{12} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} - \frac{2^{\frac{2}{3}} i}{4}$$
делаем обратную замену
$$w = z$$
$$z = w$$

Тогда, окончательный ответ:
$$z_{1} = - \frac{2^{\frac{2}{3}}}{2} - \frac{2^{\frac{2}{3}} i}{2}$$
$$z_{2} = - \frac{2^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} i}{2}$$
$$z_{3} = \frac{2^{\frac{2}{3}}}{2} - \frac{2^{\frac{2}{3}} i}{2}$$
$$z_{4} = \frac{2^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} i}{2}$$
$$z_{5} = - \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$z_{6} = - \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} - \frac{2^{\frac{2}{3}} i}{4}$$
$$z_{7} = \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$z_{8} = \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} + \frac{2^{\frac{2}{3}} i}{4}$$
$$z_{9} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4} - \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$z_{10} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} + \frac{2^{\frac{2}{3}} i}{4}$$
$$z_{11} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} i}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{4}$$
$$z_{12} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{4} - \frac{2^{\frac{2}{3}} i}{4}$$

Быстрый ответ

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

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

        2/3      2/3
       2      I*2   
z2 = - ---- + ------
        2       2   

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

      2/3      2/3
     2      I*2   
z3 = ---- - ------
      2       2   

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

      2/3      2/3
     2      I*2   
z4 = ---- + ------
      2       2   

$$z_{4} = \frac{2^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} i}{2}$$

        2/3     / 2/3    2/3   ___\    2/3   ___
       2        |2      2   *\/ 3 |   2   *\/ 3 
z5 = - ---- + I*|---- + ----------| + ----------
        4       \ 4         4     /       4     

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

        2/3     /   2/3    2/3   ___\    2/3   ___
       2        |  2      2   *\/ 3 |   2   *\/ 3 
z6 = - ---- + I*|- ---- - ----------| + ----------
        4       \   4         4     /       4     

$$z_{6} = - \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + i \left(- \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4}\right)$$

      2/3     /   2/3    2/3   ___\    2/3   ___
     2        |  2      2   *\/ 3 |   2   *\/ 3 
z7 = ---- + I*|- ---- + ----------| + ----------
      4       \   4         4     /       4     

$$z_{7} = \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + i \left(- \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4}\right)$$

      2/3     / 2/3    2/3   ___\    2/3   ___
     2        |2      2   *\/ 3 |   2   *\/ 3 
z8 = ---- + I*|---- - ----------| + ----------
      4       \ 4         4     /       4     

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

        2/3     /   2/3    2/3   ___\    2/3   ___
       2        |  2      2   *\/ 3 |   2   *\/ 3 
z9 = - ---- + I*|- ---- + ----------| - ----------
        4       \   4         4     /       4     

$$z_{9} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4} + i \left(- \frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4}\right)$$

         2/3     / 2/3    2/3   ___\    2/3   ___
        2        |2      2   *\/ 3 |   2   *\/ 3 
z10 = - ---- + I*|---- - ----------| - ----------
         4       \ 4         4     /       4     

$$z_{10} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4} + i \left(- \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}}}{4}\right)$$

       2/3     / 2/3    2/3   ___\    2/3   ___
      2        |2      2   *\/ 3 |   2   *\/ 3 
z11 = ---- + I*|---- + ----------| - ----------
       4       \ 4         4     /       4     

$$z_{11} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}}}{4} + i \left(\frac{2^{\frac{2}{3}}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3}}{4}\right)$$

       2/3     /   2/3    2/3   ___\    2/3   ___
      2        |  2      2   *\/ 3 |   2   *\/ 3 
z12 = ---- + I*|- ---- - ----------| - ----------
       4       \   4         4     /       4     

$$z_{12} = - \frac{2^{\frac{2}{3}} \sqrt{3}}{4} + \frac{2^{\frac{2}{3}}}{4} + i \left(- \frac{2^{\frac{2}{3}} \sqrt{3}}{4} - \frac{2^{\frac{2}{3}}}{4}\right)$$

Сумма и произведение корней

сумма

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

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

=

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

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

произведение

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

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

=

4

$$4$$

Численный ответ

z1 = -1.08421508149135 + 0.290514555507251*i

z2 = 1.08421508149135 - 0.290514555507251*i

z3 = -0.290514555507251 + 1.08421508149135*i

z4 = -0.7937005259841 + 0.7937005259841*i

z5 = -0.290514555507251 - 1.08421508149135*i

z6 = -1.08421508149135 - 0.290514555507251*i

z7 = 1.08421508149135 + 0.290514555507251*i

z8 = 0.290514555507251 - 1.08421508149135*i

z9 = 0.7937005259841 - 0.7937005259841*i

z10 = 0.290514555507251 + 1.08421508149135*i

z11 = -0.7937005259841 - 0.7937005259841*i

z12 = 0.7937005259841 + 0.7937005259841*i

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