x^21=400 (уравнение)

Найду корень уравнения: x^21=400

Вы ввели

 21      
x   = 400

$$x^{21} = 400$$

Подробное решение

Дано уравнение
$$x^{21} = 400$$
Т.к. степень в ур-нии равна = 21 - не содержит чётного числа в числителе, то
ур-ние будет иметь один действительный корень.
Извлечём корень 21-й степени из обеих частей ур-ния:
Получим:
$$\sqrt[21]{x^{21}} = \sqrt[21]{400}$$
или
$$x = 20^{\frac{2}{21}}$$
Раскрываем скобочки в правой части ур-ния

x = 20^2/21

Получим ответ: x = 20^(2/21)

Остальные 20 корня(ей) являются комплексными.
сделаем замену:
$$z = x$$
тогда ур-ние будет таким:
$$z^{21} = 400$$
Любое комплексное число можно представить так:
$$z = r e^{i p}$$
подставляем в уравнение
$$r^{21} e^{21 i p} = 400$$
где
$$r = 20^{\frac{2}{21}}$$
- модуль комплексного числа
Подставляем r:
$$e^{21 i p} = 1$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(21 p \right)} + \cos{\left(21 p \right)} = 1$$
значит
$$\cos{\left(21 p \right)} = 1$$
и
$$\sin{\left(21 p \right)} = 0$$
тогда
$$p = \frac{2 \pi N}{21}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для z
Значит, решением будет для z:
$$z_{1} = 20^{\frac{2}{21}}$$
$$z_{2} = - \frac{20^{\frac{2}{21}}}{2} - \frac{20^{\frac{2}{21}} \sqrt{3} i}{2}$$
$$z_{3} = - \frac{20^{\frac{2}{21}}}{2} + \frac{20^{\frac{2}{21}} \sqrt{3} i}{2}$$
$$z_{4} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{21} \right)}$$
$$z_{5} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{21} \right)}$$
$$z_{6} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{21} \right)}$$
$$z_{7} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{21} \right)}$$
$$z_{8} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{7} \right)}$$
$$z_{9} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{7} \right)}$$
$$z_{10} = 20^{\frac{2}{21}} \cos{\left(\frac{4 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{4 \pi}{21} \right)}$$
$$z_{11} = 20^{\frac{2}{21}} \cos{\left(\frac{4 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{4 \pi}{21} \right)}$$
$$z_{12} = - 20^{\frac{2}{21}} \cos{\left(\frac{5 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{5 \pi}{21} \right)}$$
$$z_{13} = - 20^{\frac{2}{21}} \cos{\left(\frac{5 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{5 \pi}{21} \right)}$$
$$z_{14} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{7} \right)}$$
$$z_{15} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{7} \right)}$$
$$z_{16} = 20^{\frac{2}{21}} \cos{\left(\frac{8 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{8 \pi}{21} \right)}$$
$$z_{17} = 20^{\frac{2}{21}} \cos{\left(\frac{8 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{8 \pi}{21} \right)}$$
$$z_{18} = - 20^{\frac{2}{21}} \cos{\left(\frac{3 \pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{3 \pi}{7} \right)}$$
$$z_{19} = - 20^{\frac{2}{21}} \cos{\left(\frac{3 \pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{3 \pi}{7} \right)}$$
$$z_{20} = 20^{\frac{2}{21}} \cos{\left(\frac{10 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{10 \pi}{21} \right)}$$
$$z_{21} = 20^{\frac{2}{21}} \cos{\left(\frac{10 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{10 \pi}{21} \right)}$$
делаем обратную замену
$$z = x$$
$$x = z$$

Тогда, окончательный ответ:
$$x_{1} = 20^{\frac{2}{21}}$$
$$x_{2} = - \frac{20^{\frac{2}{21}}}{2} - \frac{20^{\frac{2}{21}} \sqrt{3} i}{2}$$
$$x_{3} = - \frac{20^{\frac{2}{21}}}{2} + \frac{20^{\frac{2}{21}} \sqrt{3} i}{2}$$
$$x_{4} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{21} \right)}$$
$$x_{5} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{21} \right)}$$
$$x_{6} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{21} \right)}$$
$$x_{7} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{21} \right)}$$
$$x_{8} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{7} \right)}$$
$$x_{9} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{7} \right)}$$
$$x_{10} = 20^{\frac{2}{21}} \cos{\left(\frac{4 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{4 \pi}{21} \right)}$$
$$x_{11} = 20^{\frac{2}{21}} \cos{\left(\frac{4 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{4 \pi}{21} \right)}$$
$$x_{12} = - 20^{\frac{2}{21}} \cos{\left(\frac{5 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{5 \pi}{21} \right)}$$
$$x_{13} = - 20^{\frac{2}{21}} \cos{\left(\frac{5 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{5 \pi}{21} \right)}$$
$$x_{14} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{7} \right)}$$
$$x_{15} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{7} \right)}$$
$$x_{16} = 20^{\frac{2}{21}} \cos{\left(\frac{8 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{8 \pi}{21} \right)}$$
$$x_{17} = 20^{\frac{2}{21}} \cos{\left(\frac{8 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{8 \pi}{21} \right)}$$
$$x_{18} = - 20^{\frac{2}{21}} \cos{\left(\frac{3 \pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{3 \pi}{7} \right)}$$
$$x_{19} = - 20^{\frac{2}{21}} \cos{\left(\frac{3 \pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{3 \pi}{7} \right)}$$
$$x_{20} = 20^{\frac{2}{21}} \cos{\left(\frac{10 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{10 \pi}{21} \right)}$$
$$x_{21} = 20^{\frac{2}{21}} \cos{\left(\frac{10 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{10 \pi}{21} \right)}$$

Быстрый ответ

       2/21
x1 = 20    

$$x_{1} = 20^{\frac{2}{21}}$$

         2/21       ___   2/21
       20       I*\/ 3 *20    
x2 = - ------ - --------------
         2            2       

$$x_{2} = - \frac{20^{\frac{2}{21}}}{2} - \frac{20^{\frac{2}{21}} \sqrt{3} i}{2}$$

         2/21       ___   2/21
       20       I*\/ 3 *20    
x3 = - ------ + --------------
         2            2       

$$x_{3} = - \frac{20^{\frac{2}{21}}}{2} + \frac{20^{\frac{2}{21}} \sqrt{3} i}{2}$$

         2/21    /pi\       2/21    /pi\
x4 = - 20    *cos|--| - I*20    *sin|--|
                 \21/               \21/

$$x_{4} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{21} \right)}$$

         2/21    /pi\       2/21    /pi\
x5 = - 20    *cos|--| + I*20    *sin|--|
                 \21/               \21/

$$x_{5} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{21} \right)}$$

       2/21    /2*pi\       2/21    /2*pi\
x6 = 20    *cos|----| - I*20    *sin|----|
               \ 21 /               \ 21 /

$$x_{6} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{21} \right)}$$

       2/21    /2*pi\       2/21    /2*pi\
x7 = 20    *cos|----| + I*20    *sin|----|
               \ 21 /               \ 21 /

$$x_{7} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{21} \right)}$$

         2/21    /pi\       2/21    /pi\
x8 = - 20    *cos|--| - I*20    *sin|--|
                 \7 /               \7 /

$$x_{8} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{7} \right)}$$

         2/21    /pi\       2/21    /pi\
x9 = - 20    *cos|--| + I*20    *sin|--|
                 \7 /               \7 /

$$x_{9} = - 20^{\frac{2}{21}} \cos{\left(\frac{\pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{\pi}{7} \right)}$$

        2/21    /4*pi\       2/21    /4*pi\
x10 = 20    *cos|----| - I*20    *sin|----|
                \ 21 /               \ 21 /

$$x_{10} = 20^{\frac{2}{21}} \cos{\left(\frac{4 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{4 \pi}{21} \right)}$$

        2/21    /4*pi\       2/21    /4*pi\
x11 = 20    *cos|----| + I*20    *sin|----|
                \ 21 /               \ 21 /

$$x_{11} = 20^{\frac{2}{21}} \cos{\left(\frac{4 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{4 \pi}{21} \right)}$$

          2/21    /5*pi\       2/21    /5*pi\
x12 = - 20    *cos|----| - I*20    *sin|----|
                  \ 21 /               \ 21 /

$$x_{12} = - 20^{\frac{2}{21}} \cos{\left(\frac{5 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{5 \pi}{21} \right)}$$

          2/21    /5*pi\       2/21    /5*pi\
x13 = - 20    *cos|----| + I*20    *sin|----|
                  \ 21 /               \ 21 /

$$x_{13} = - 20^{\frac{2}{21}} \cos{\left(\frac{5 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{5 \pi}{21} \right)}$$

        2/21    /2*pi\       2/21    /2*pi\
x14 = 20    *cos|----| - I*20    *sin|----|
                \ 7  /               \ 7  /

$$x_{14} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{7} \right)}$$

        2/21    /2*pi\       2/21    /2*pi\
x15 = 20    *cos|----| + I*20    *sin|----|
                \ 7  /               \ 7  /

$$x_{15} = 20^{\frac{2}{21}} \cos{\left(\frac{2 \pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{2 \pi}{7} \right)}$$

        2/21    /8*pi\       2/21    /8*pi\
x16 = 20    *cos|----| - I*20    *sin|----|
                \ 21 /               \ 21 /

$$x_{16} = 20^{\frac{2}{21}} \cos{\left(\frac{8 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{8 \pi}{21} \right)}$$

        2/21    /8*pi\       2/21    /8*pi\
x17 = 20    *cos|----| + I*20    *sin|----|
                \ 21 /               \ 21 /

$$x_{17} = 20^{\frac{2}{21}} \cos{\left(\frac{8 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{8 \pi}{21} \right)}$$

          2/21    /3*pi\       2/21    /3*pi\
x18 = - 20    *cos|----| - I*20    *sin|----|
                  \ 7  /               \ 7  /

$$x_{18} = - 20^{\frac{2}{21}} \cos{\left(\frac{3 \pi}{7} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{3 \pi}{7} \right)}$$

          2/21    /3*pi\       2/21    /3*pi\
x19 = - 20    *cos|----| + I*20    *sin|----|
                  \ 7  /               \ 7  /

$$x_{19} = - 20^{\frac{2}{21}} \cos{\left(\frac{3 \pi}{7} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{3 \pi}{7} \right)}$$

        2/21    /10*pi\       2/21    /10*pi\
x20 = 20    *cos|-----| - I*20    *sin|-----|
                \  21 /               \  21 /

$$x_{20} = 20^{\frac{2}{21}} \cos{\left(\frac{10 \pi}{21} \right)} - 20^{\frac{2}{21}} i \sin{\left(\frac{10 \pi}{21} \right)}$$

        2/21    /10*pi\       2/21    /10*pi\
x21 = 20    *cos|-----| + I*20    *sin|-----|
                \  21 /               \  21 /

$$x_{21} = 20^{\frac{2}{21}} \cos{\left(\frac{10 \pi}{21} \right)} + 20^{\frac{2}{21}} i \sin{\left(\frac{10 \pi}{21} \right)}$$

Численный ответ

x1 = -0.295990990971579 - 1.2968212640578*i

x2 = 0.0994038361369948 + 1.32645201006673*i

x3 = 0.829348327061535 + 1.03996990824208*i

x4 = 0.0994038361369948 - 1.32645201006673*i

x5 = -0.665085719597258 + 1.15196225773096*i

x6 = 1.33017143919452

x7 = 0.485966196178246 + 1.23822167394246*i

x8 = -0.975084663356114 + 0.904746349501618*i

x9 = -0.665085719597258 - 1.15196225773096*i

x10 = -1.31531452323978 + 0.198251765700374*i

x11 = -1.19844305568721 - 0.5771397577051*i

x12 = 1.27107565432769 + 0.392074914556183*i

x13 = -0.295990990971579 + 1.2968212640578*i

x14 = 1.27107565432769 - 0.392074914556183*i

x15 = 0.829348327061535 - 1.03996990824208*i

x16 = 1.09903921955022 + 0.749312252361626*i

x17 = -0.975084663356114 - 0.904746349501618*i

x18 = 0.485966196178246 - 1.23822167394246*i

x19 = -1.19844305568721 + 0.5771397577051*i

x20 = 1.09903921955022 - 0.749312252361626*i

x21 = -1.31531452323978 - 0.198251765700374*i

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