x^9=-512 (уравнение)

Найду корень уравнения: x^9=-512

Вы ввели

 9       
x  = -512

$$x^{9} = -512$$

Подробное решение

Дано уравнение
$$x^{9} = -512$$
Т.к. степень в ур-нии равна = 9 - не содержит чётного числа в числителе, то
ур-ние будет иметь один действительный корень.
Извлечём корень 9-й степени из обеих частей ур-ния:
Получим:
$$\sqrt[9]{\left(1 x + 0\right)^{9}} = \sqrt[9]{-512}$$
или
$$x = 2 \sqrt[9]{-1}$$
Раскрываем скобочки в правой части ур-ния

x = -2*1^1/9

Получим ответ: x = 2*(-1)^(1/9)

Остальные 8 корня(ей) являются комплексными.
сделаем замену:
$$z = x$$
тогда ур-ние будет таким:
$$z^{9} = -512$$
Любое комплексное число можно представить так:
$$z = r e^{i p}$$
подставляем в уравнение
$$r^{9} e^{9 i p} = -512$$
где
$$r = 2$$
- модуль комплексного числа
Подставляем r:
$$e^{9 i p} = -1$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(9 p \right)} + \cos{\left(9 p \right)} = -1$$
значит
$$\cos{\left(9 p \right)} = -1$$
и
$$\sin{\left(9 p \right)} = 0$$
тогда
$$p = \frac{2 \pi N}{9} + \frac{\pi}{9}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для z
Значит, решением будет для z:
$$z_{1} = 2 \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)}$$
$$z_{2} = - 2 \cos^{2}{\left(\frac{\pi}{9} \right)} - 2 \sin^{2}{\left(\frac{\pi}{9} \right)}$$
$$z_{3} = - 2 \cos^{2}{\left(\frac{\pi}{9} \right)} + 2 \sin^{2}{\left(\frac{\pi}{9} \right)} - 4 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$z_{4} = - 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$z_{5} = 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} - 2 i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}$$
$$z_{6} = - 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$z_{7} = 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} - 2 i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}$$
$$z_{8} = - \cos{\left(\frac{\pi}{9} \right)} - \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} - i \sin{\left(\frac{\pi}{9} \right)} + \sqrt{3} i \cos{\left(\frac{\pi}{9} \right)}$$
$$z_{9} = - \cos{\left(\frac{\pi}{9} \right)} + \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} - \sqrt{3} i \cos{\left(\frac{\pi}{9} \right)} - i \sin{\left(\frac{\pi}{9} \right)}$$
делаем обратную замену
$$z = x$$
$$x = z$$

Тогда, окончательный ответ:
$$x_{1} = 2 \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)}$$
$$x_{2} = - 2 \cos^{2}{\left(\frac{\pi}{9} \right)} - 2 \sin^{2}{\left(\frac{\pi}{9} \right)}$$
$$x_{3} = - 2 \cos^{2}{\left(\frac{\pi}{9} \right)} + 2 \sin^{2}{\left(\frac{\pi}{9} \right)} - 4 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$x_{4} = - 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$x_{5} = 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} - 2 i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}$$
$$x_{6} = - 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$x_{7} = 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} - 2 i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}$$
$$x_{8} = - \cos{\left(\frac{\pi}{9} \right)} - \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} - i \sin{\left(\frac{\pi}{9} \right)} + \sqrt{3} i \cos{\left(\frac{\pi}{9} \right)}$$
$$x_{9} = - \cos{\left(\frac{\pi}{9} \right)} + \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} - \sqrt{3} i \cos{\left(\frac{\pi}{9} \right)} - i \sin{\left(\frac{\pi}{9} \right)}$$

Быстрый ответ

            2/pi\        2/pi\
x1 = - 2*cos |--| - 2*sin |--|
             \9 /         \9 /

$$x_{1} = - 2 \cos^{2}{\left(\frac{\pi}{9} \right)} - 2 \sin^{2}{\left(\frac{\pi}{9} \right)}$$

       /     /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\
x2 = I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----|
       \     \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /

$$x_{2} = - 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

       /       /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\
x3 = I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----|
       \       \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /

$$x_{3} = 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(- 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right)$$

       /     /pi\    /4*pi\        /4*pi\    /pi\\        /pi\    /4*pi\        /pi\    /4*pi\
x4 = I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----|
       \     \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /

$$x_{4} = - 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

       /       /pi\    /4*pi\        /4*pi\    /pi\\        /pi\    /4*pi\        /pi\    /4*pi\
x5 = I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----|
       \       \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /

$$x_{5} = 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + i \left(- 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right)$$

          /pi\     /     /pi\     ___    /pi\\     ___    /pi\
x6 = - cos|--| + I*|- sin|--| + \/ 3 *cos|--|| - \/ 3 *sin|--|
          \9 /     \     \9 /            \9 //            \9 /

$$x_{6} = - \cos{\left(\frac{\pi}{9} \right)} - \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + i \left(- \sin{\left(\frac{\pi}{9} \right)} + \sqrt{3} \cos{\left(\frac{\pi}{9} \right)}\right)$$

          /pi\     /     /pi\     ___    /pi\\     ___    /pi\
x7 = - cos|--| + I*|- sin|--| - \/ 3 *cos|--|| + \/ 3 *sin|--|
          \9 /     \     \9 /            \9 //            \9 /

$$x_{7} = - \cos{\left(\frac{\pi}{9} \right)} + \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + i \left(- \sqrt{3} \cos{\left(\frac{\pi}{9} \right)} - \sin{\left(\frac{\pi}{9} \right)}\right)$$

          /pi\          /pi\
x8 = 2*cos|--| + 2*I*sin|--|
          \9 /          \9 /

$$x_{8} = 2 \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)}$$

            2/pi\        2/pi\          /pi\    /pi\
x9 = - 2*cos |--| + 2*sin |--| - 4*I*cos|--|*sin|--|
             \9 /         \9 /          \9 /    \9 /

$$x_{9} = - 2 \cos^{2}{\left(\frac{\pi}{9} \right)} + 2 \sin^{2}{\left(\frac{\pi}{9} \right)} - 4 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$

Сумма и произведение корней

сумма

           2/pi\        2/pi\     /     /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\     /       /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\     /     /pi\    /4*pi\        /4*pi\    /pi\\        /pi\    /4*pi\        /pi\    /4*pi\     /       /pi\    /4*pi\        /4*pi\    /pi\\        /pi\    /4*pi\        /pi\    /4*pi\        /pi\     /     /pi\     ___    /pi\\     ___    /pi\        /pi\     /     /pi\     ___    /pi\\     ___    /pi\        /pi\          /pi\          2/pi\        2/pi\          /pi\    /pi\
0 + - 2*cos |--| - 2*sin |--| + I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----| + I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----| + I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----| + I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----| + - cos|--| + I*|- sin|--| + \/ 3 *cos|--|| - \/ 3 *sin|--| + - cos|--| + I*|- sin|--| - \/ 3 *cos|--|| + \/ 3 *sin|--| + 2*cos|--| + 2*I*sin|--| + - 2*cos |--| + 2*sin |--| - 4*I*cos|--|*sin|--|
            \9 /         \9 /     \     \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /     \       \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /     \     \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /     \       \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  /        \9 /     \     \9 /            \9 //            \9 /        \9 /     \     \9 /            \9 //            \9 /        \9 /          \9 /           \9 /         \9 /          \9 /    \9 /

$$\left(- 2 \cos^{2}{\left(\frac{\pi}{9} \right)} + 2 \sin^{2}{\left(\frac{\pi}{9} \right)} - 4 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) - \left(- 4 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} - 4 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin^{2}{\left(\frac{\pi}{9} \right)} + 2 \cos^{2}{\left(\frac{\pi}{9} \right)} - i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) - i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) - i \left(- \sin{\left(\frac{\pi}{9} \right)} + \sqrt{3} \cos{\left(\frac{\pi}{9} \right)}\right) - 2 i \sin{\left(\frac{\pi}{9} \right)} - i \left(- 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right) - i \left(- 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right) - i \left(- \sqrt{3} \cos{\left(\frac{\pi}{9} \right)} - \sin{\left(\frac{\pi}{9} \right)}\right)\right)$$

=

       2/pi\     /     /pi\     ___    /pi\\     /     /pi\     ___    /pi\\     /       /pi\    /2*pi\        /2*pi\    /pi\\     /       /pi\    /4*pi\        /4*pi\    /pi\\     /     /pi\    /2*pi\        /2*pi\    /pi\\     /     /pi\    /4*pi\        /4*pi\    /pi\\          /pi\        /pi\    /2*pi\        /pi\    /4*pi\          /pi\    /pi\
- 4*cos |--| + I*|- sin|--| + \/ 3 *cos|--|| + I*|- sin|--| - \/ 3 *cos|--|| + I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*I*sin|--| + 4*cos|--|*cos|----| + 4*cos|--|*cos|----| - 4*I*cos|--|*sin|--|
        \9 /     \     \9 /            \9 //     \     \9 /            \9 //     \       \9 /    \ 9  /        \ 9  /    \9 //     \       \9 /    \ 9  /        \ 9  /    \9 //     \     \9 /    \ 9  /        \ 9  /    \9 //     \     \9 /    \ 9  /        \ 9  /    \9 //          \9 /        \9 /    \ 9  /        \9 /    \ 9  /          \9 /    \9 /

$$- 4 \cos^{2}{\left(\frac{\pi}{9} \right)} + 4 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 4 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(- \sqrt{3} \cos{\left(\frac{\pi}{9} \right)} - \sin{\left(\frac{\pi}{9} \right)}\right) + i \left(- 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right) - 4 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + i \left(- 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right) + 2 i \sin{\left(\frac{\pi}{9} \right)} + i \left(- \sin{\left(\frac{\pi}{9} \right)} + \sqrt{3} \cos{\left(\frac{\pi}{9} \right)}\right) + i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) + i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

произведение

  /       2/pi\        2/pi\\ /  /     /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\\ /  /       /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\\ /  /     /pi\    /4*pi\        /4*pi\    /pi\\        /pi\    /4*pi\        /pi\    /4*pi\\ /  /       /pi\    /4*pi\        /4*pi\    /pi\\        /pi\    /4*pi\        /pi\    /4*pi\\ /     /pi\     /     /pi\     ___    /pi\\     ___    /pi\\ /     /pi\     /     /pi\     ___    /pi\\     ___    /pi\\ /     /pi\          /pi\\ /       2/pi\        2/pi\          /pi\    /pi\\
1*|- 2*cos |--| - 2*sin |--||*|I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----||*|I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----||*|I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----||*|I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----||*|- cos|--| + I*|- sin|--| + \/ 3 *cos|--|| - \/ 3 *sin|--||*|- cos|--| + I*|- sin|--| - \/ 3 *cos|--|| + \/ 3 *sin|--||*|2*cos|--| + 2*I*sin|--||*|- 2*cos |--| + 2*sin |--| - 4*I*cos|--|*sin|--||
  \        \9 /         \9 // \  \     \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  // \  \       \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  // \  \     \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  // \  \       \9 /    \ 9  /        \ 9  /    \9 //        \9 /    \ 9  /        \9 /    \ 9  // \     \9 /     \     \9 /            \9 //            \9 // \     \9 /     \     \9 /            \9 //            \9 // \     \9 /          \9 // \        \9 /         \9 /          \9 /    \9 //

$$1 \left(- 2 \cos^{2}{\left(\frac{\pi}{9} \right)} - 2 \sin^{2}{\left(\frac{\pi}{9} \right)}\right) \left(- 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(- 2 \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right)\right) \left(- 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \left(2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + i \left(- 2 \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + 2 \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right)\right) \left(- \cos{\left(\frac{\pi}{9} \right)} - \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + i \left(- \sin{\left(\frac{\pi}{9} \right)} + \sqrt{3} \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(- \cos{\left(\frac{\pi}{9} \right)} + \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + i \left(- \sqrt{3} \cos{\left(\frac{\pi}{9} \right)} - \sin{\left(\frac{\pi}{9} \right)}\right)\right) \left(2 \cos{\left(\frac{\pi}{9} \right)} + 2 i \sin{\left(\frac{\pi}{9} \right)}\right) \left(- 2 \cos^{2}{\left(\frac{\pi}{9} \right)} + 2 \sin^{2}{\left(\frac{\pi}{9} \right)} - 4 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

=

/      /   /pi\                          \\                     /              /   /pi\                          \\                                                    2*pi*I
|      |cos|--|     ___                  ||                     |              |cos|--|     ___                  ||                                                    ------
|1     |   \18/   \/ 3       /pi\    /pi\|| /              ___\ |     /pi\     |   \18/   \/ 3       /pi\    /pi\|| /     /pi\      /pi\\ /       /2*pi\      /5*pi\\    9   
|- + I*|------- - ----- - cos|--|*cos|--|||*\256 + 256*I*\/ 3 /*|- sin|--| + I*|------- - ----- + cos|--|*cos|--|||*|I*cos|--| + sin|--||*|- I*sin|----| + sin|----||*e      
\2     \   2        4        \9 /    \18///                     \     \18/     \   2        4        \9 /    \18/// \     \18/      \18// \       \ 9  /      \ 18 //        

$$\left(\frac{1}{2} + i \left(- \cos{\left(\frac{\pi}{18} \right)} \cos{\left(\frac{\pi}{9} \right)} - \frac{\sqrt{3}}{4} + \frac{\cos{\left(\frac{\pi}{18} \right)}}{2}\right)\right) \left(256 + 256 \sqrt{3} i\right) \left(- \sin{\left(\frac{\pi}{18} \right)} + i \left(- \frac{\sqrt{3}}{4} + \frac{\cos{\left(\frac{\pi}{18} \right)}}{2} + \cos{\left(\frac{\pi}{18} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(\sin{\left(\frac{\pi}{18} \right)} + i \cos{\left(\frac{\pi}{18} \right)}\right) \left(\sin{\left(\frac{5 \pi}{18} \right)} - i \sin{\left(\frac{2 \pi}{9} \right)}\right) e^{\frac{2 i \pi}{9}}$$

Численный ответ

x1 = -0.347296355333861 + 1.96961550602442*i

x2 = -1.53208888623796 + 1.28557521937308*i

x3 = 1.87938524157182 - 0.684040286651337*i

x4 = -1.53208888623796 - 1.28557521937308*i

x5 = -2.0

x6 = 1.87938524157182 + 0.684040286651337*i

x7 = -0.347296355333861 - 1.96961550602442*i

x8 = 1.0 + 1.73205080756888*i

x9 = 1.0 - 1.73205080756888*i

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