x^9=-512

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

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:
$$sqrt[9]{x^{9}} = sqrt[9]{-512}$$
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
$$p = frac{2 pi}{9} N + frac{pi}{9}$$
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:
$$x_{1} = 2 cos{left (frac{pi}{9} right )} + 2 i sin{left (frac{pi}{9} right )}$$
$$x_{2} = – 2 cos^{2}{left (frac{pi}{9} right )} – 2 sin^{2}{left (frac{pi}{9} right )}$$
$$x_{3} = – 2 cos^{2}{left (frac{pi}{9} right )} + 2 sin^{2}{left (frac{pi}{9} right )} – 4 i sin{left (frac{pi}{9} right )} cos{left (frac{pi}{9} right )}$$
$$x_{4} = – 2 sin{left (frac{pi}{9} right )} sin{left (frac{2 pi}{9} right )} + 2 cos{left (frac{pi}{9} right )} cos{left (frac{2 pi}{9} right )} + 2 i sin{left (frac{pi}{9} right )} cos{left (frac{2 pi}{9} right )} + 2 i sin{left (frac{2 pi}{9} right )} cos{left (frac{pi}{9} right )}$$
$$x_{5} = 2 sin{left (frac{pi}{9} right )} sin{left (frac{2 pi}{9} right )} + 2 cos{left (frac{pi}{9} right )} cos{left (frac{2 pi}{9} right )} – 2 i sin{left (frac{2 pi}{9} right )} cos{left (frac{pi}{9} right )} + 2 i sin{left (frac{pi}{9} right )} cos{left (frac{2 pi}{9} right )}$$
$$x_{6} = – 2 sin{left (frac{pi}{9} right )} sin{left (frac{4 pi}{9} right )} + 2 cos{left (frac{pi}{9} right )} cos{left (frac{4 pi}{9} right )} + 2 i sin{left (frac{pi}{9} right )} cos{left (frac{4 pi}{9} right )} + 2 i sin{left (frac{4 pi}{9} right )} cos{left (frac{pi}{9} right )}$$
$$x_{7} = 2 cos{left (frac{pi}{9} right )} cos{left (frac{4 pi}{9} right )} + 2 sin{left (frac{pi}{9} right )} sin{left (frac{4 pi}{9} right )} – 2 i sin{left (frac{4 pi}{9} right )} cos{left (frac{pi}{9} right )} + 2 i sin{left (frac{pi}{9} right )} cos{left (frac{4 pi}{9} right )}$$
$$x_{8} = – cos{left (frac{pi}{9} right )} – sqrt{3} sin{left (frac{pi}{9} right )} – i sin{left (frac{pi}{9} right )} + sqrt{3} i cos{left (frac{pi}{9} right )}$$
$$x_{9} = – cos{left (frac{pi}{9} right )} + sqrt{3} sin{left (frac{pi}{9} right )} – sqrt{3} i cos{left (frac{pi}{9} right )} – i sin{left (frac{pi}{9} right )}$$

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 /

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

x1 = -1.53208888624 – 1.28557521937*i

x2 = -0.347296355334 – 1.96961550602*i

x3 = -2.00000000000000

x4 = -0.347296355334 + 1.96961550602*i

x5 = -1.53208888624 + 1.28557521937*i

x6 = 1.0 + 1.73205080757*i

x7 = 1.87938524157 + 0.684040286651*i

x8 = 1.0 – 1.73205080757*i

x9 = 1.87938524157 – 0.684040286651*i