x^6=-1/x (уравнение)

Найду корень уравнения: x^6=-1/x

Вы ввели

 6   -1 
x  = ---
      x 

$$x^{6} = - \frac{1}{x}$$

Подробное решение

Дано уравнение
$$x^{6} = - \frac{1}{x}$$
преобразуем
$$\frac{1}{x^{7}} = -1$$
Т.к. степень в ур-нии равна = -7 - не содержит чётного числа в числителе, то
ур-ние будет иметь один действительный корень.
Извлечём корень -7-й степени из обеих частей ур-ния:
Получим:
$$\frac{1}{\sqrt[7]{\frac{1}{\left(1 x + 0\right)^{7}}}} = \frac{1}{\sqrt[7]{-1}}$$
или
$$x = - \left(-1\right)^{\frac{6}{7}}$$
Раскрываем скобочки в правой части ур-ния

x = 1^6/7

Получим ответ: x = -(-1)^(6/7)

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

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

Быстрый ответ

          2/pi\      2/pi\
x1 = - cos |--| - sin |--|
           \7 /       \7 /

$$x_{1} = - \cos^{2}{\left(\frac{\pi}{7} \right)} - \sin^{2}{\left(\frac{\pi}{7} \right)}$$

       /   /pi\    /2*pi\      /2*pi\    /pi\\      /pi\    /2*pi\      /pi\    /2*pi\
x2 = I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----|
       \   \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  /

$$x_{2} = - \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(\sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

       /   /2*pi\    /pi\      /pi\    /2*pi\\      /pi\    /2*pi\      /pi\    /2*pi\
x3 = I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----|
       \   \ 7  /    \7 /      \7 /    \ 7  //      \7 /    \ 7  /      \7 /    \ 7  /

$$x_{3} = \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(- \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right)$$

       /   /pi\    /3*pi\      /3*pi\    /pi\\      /pi\    /3*pi\      /pi\    /3*pi\
x4 = I*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----|
       \   \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  /

$$x_{4} = - \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + i \left(- \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

       /     /pi\    /3*pi\      /3*pi\    /pi\\      /pi\    /3*pi\      /pi\    /3*pi\
x5 = I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----|
       \     \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  /

$$x_{5} = - \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} + i \left(- \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)$$

          /pi\      /pi\
x6 = I*sin|--| + cos|--|
          \7 /      \7 /

$$x_{6} = \cos{\left(\frac{\pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)}$$

        2/pi\      2/pi\          /pi\    /pi\
x7 = sin |--| - cos |--| - 2*I*cos|--|*sin|--|
         \7 /       \7 /          \7 /    \7 /

$$x_{7} = - \cos^{2}{\left(\frac{\pi}{7} \right)} + \sin^{2}{\left(\frac{\pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$

Сумма и произведение корней

сумма

         2/pi\      2/pi\     /   /pi\    /2*pi\      /2*pi\    /pi\\      /pi\    /2*pi\      /pi\    /2*pi\     /   /2*pi\    /pi\      /pi\    /2*pi\\      /pi\    /2*pi\      /pi\    /2*pi\     /   /pi\    /3*pi\      /3*pi\    /pi\\      /pi\    /3*pi\      /pi\    /3*pi\     /     /pi\    /3*pi\      /3*pi\    /pi\\      /pi\    /3*pi\      /pi\    /3*pi\        /pi\      /pi\      2/pi\      2/pi\          /pi\    /pi\
0 + - cos |--| - sin |--| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----| + I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----| + I*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----| + I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----| + I*sin|--| + cos|--| + sin |--| - cos |--| - 2*I*cos|--|*sin|--|
          \7 /       \7 /     \   \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  /     \   \ 7  /    \7 /      \7 /    \ 7  //      \7 /    \ 7  /      \7 /    \ 7  /     \   \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  /     \     \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  /        \7 /      \7 /       \7 /       \7 /          \7 /    \7 /

$$\left(- \cos^{2}{\left(\frac{\pi}{7} \right)} + \sin^{2}{\left(\frac{\pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right) - \left(- 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} - \cos{\left(\frac{\pi}{7} \right)} + \sin^{2}{\left(\frac{\pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \cos^{2}{\left(\frac{\pi}{7} \right)} - i \left(\sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right) - i \left(- \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right) - i \sin{\left(\frac{\pi}{7} \right)} - i \left(- \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right) - i \left(- \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)\right)$$

=

       2/pi\     /   /pi\    /2*pi\      /2*pi\    /pi\\     /   /pi\    /3*pi\      /3*pi\    /pi\\     /   /2*pi\    /pi\      /pi\    /2*pi\\     /     /pi\    /3*pi\      /3*pi\    /pi\\        /pi\        /pi\    /3*pi\        /pi\    /2*pi\          /pi\    /pi\      /pi\
- 2*cos |--| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + I*|cos|--|*sin|----| - cos|----|*sin|--|| + I*|cos|----|*sin|--| - cos|--|*sin|----|| + I*|- cos|--|*sin|----| - cos|----|*sin|--|| + I*sin|--| - 2*cos|--|*cos|----| + 2*cos|--|*cos|----| - 2*I*cos|--|*sin|--| + cos|--|
        \7 /     \   \7 /    \ 7  /      \ 7  /    \7 //     \   \7 /    \ 7  /      \ 7  /    \7 //     \   \ 7  /    \7 /      \7 /    \ 7  //     \     \7 /    \ 7  /      \ 7  /    \7 //        \7 /        \7 /    \ 7  /        \7 /    \ 7  /          \7 /    \7 /      \7 /

$$- 2 \cos^{2}{\left(\frac{\pi}{7} \right)} - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(- \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right) - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + i \left(- \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right) + i \sin{\left(\frac{\pi}{7} \right)} + i \left(- \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right) + i \left(\sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

произведение

  /     2/pi\      2/pi\\ /  /   /pi\    /2*pi\      /2*pi\    /pi\\      /pi\    /2*pi\      /pi\    /2*pi\\ /  /   /2*pi\    /pi\      /pi\    /2*pi\\      /pi\    /2*pi\      /pi\    /2*pi\\ /  /   /pi\    /3*pi\      /3*pi\    /pi\\      /pi\    /3*pi\      /pi\    /3*pi\\ /  /     /pi\    /3*pi\      /3*pi\    /pi\\      /pi\    /3*pi\      /pi\    /3*pi\\ /     /pi\      /pi\\ /   2/pi\      2/pi\          /pi\    /pi\\
1*|- cos |--| - sin |--||*|I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----||*|I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----||*|I*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----||*|I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----||*|I*sin|--| + cos|--||*|sin |--| - cos |--| - 2*I*cos|--|*sin|--||
  \      \7 /       \7 // \  \   \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  // \  \   \ 7  /    \7 /      \7 /    \ 7  //      \7 /    \ 7  /      \7 /    \ 7  // \  \   \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  // \  \     \7 /    \ 7  /      \ 7  /    \7 //      \7 /    \ 7  /      \7 /    \ 7  // \     \7 /      \7 // \    \7 /       \7 /          \7 /    \7 //

$$1 \left(- \cos^{2}{\left(\frac{\pi}{7} \right)} - \sin^{2}{\left(\frac{\pi}{7} \right)}\right) \left(- \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(\sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(\sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(- \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right)\right) \left(- \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + i \left(- \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(- \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} + i \left(- \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)\right) \left(\cos{\left(\frac{\pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)}\right) \left(- \cos^{2}{\left(\frac{\pi}{7} \right)} + \sin^{2}{\left(\frac{\pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

=

-1

$$-1$$

Численный ответ

x1 = 0.900968867902419 + 0.433883739117558*i

x2 = -1.0

x3 = -0.623489801858733 - 0.78183148246803*i

x4 = 0.222520933956314 + 0.974927912181824*i

x5 = 0.900968867902419 - 0.433883739117558*i

x6 = -0.623489801858733 + 0.78183148246803*i

x7 = 0.222520933956314 - 0.974927912181824*i

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