Решите уравнение y^7=f (у в степени 7 равно f) - Найдите корень уравнения подробно по-шагам. [Есть ОТВЕТ!]

y^7=f (уравнение)

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    Найду корень уравнения: y^7=f

    Решение

    Вы ввели [src]
     7    
    y  = f
    $$y^{7} = f$$
    Быстрый ответ [src]
            _________________                                 _________________                         
         14/   2        2        /atan2(im(f), re(f))\     14/   2        2        /atan2(im(f), re(f))\
    y1 = \/  im (f) + re (f) *cos|-------------------| + I*\/  im (f) + re (f) *sin|-------------------|
                                 \         7         /                             \         7         /
    $$y_{1} = i \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}$$
           /     _________________                                       _________________                                 \      _________________                                       _________________                                 
           |  14/   2        2        /pi\    /atan2(im(f), re(f))\   14/   2        2        /atan2(im(f), re(f))\    /pi\|   14/   2        2        /pi\    /atan2(im(f), re(f))\   14/   2        2        /pi\    /atan2(im(f), re(f))\
    y2 = I*|- \/  im (f) + re (f) *cos|--|*sin|-------------------| - \/  im (f) + re (f) *cos|-------------------|*sin|--|| + \/  im (f) + re (f) *sin|--|*sin|-------------------| - \/  im (f) + re (f) *cos|--|*cos|-------------------|
           \                          \7 /    \         7         /                           \         7         /    \7 //                           \7 /    \         7         /                           \7 /    \         7         /
    $$y_{2} = i \left(- \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} \cos{\left (\frac{\pi}{7} \right )} - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{\pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}\right) + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{\pi}{7} \right )} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \cos{\left (\frac{\pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}$$
           /   _________________                                       _________________                                 \      _________________                                       _________________                                 
           |14/   2        2        /atan2(im(f), re(f))\    /pi\   14/   2        2        /pi\    /atan2(im(f), re(f))\|   14/   2        2        /pi\    /atan2(im(f), re(f))\   14/   2        2        /pi\    /atan2(im(f), re(f))\
    y3 = I*|\/  im (f) + re (f) *cos|-------------------|*sin|--| - \/  im (f) + re (f) *cos|--|*sin|-------------------|| - \/  im (f) + re (f) *cos|--|*cos|-------------------| - \/  im (f) + re (f) *sin|--|*sin|-------------------|
           \                        \         7         /    \7 /                           \7 /    \         7         //                           \7 /    \         7         /                           \7 /    \         7         /
    $$y_{3} = i \left(- \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} \cos{\left (\frac{\pi}{7} \right )} + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{\pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}\right) - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{\pi}{7} \right )} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \cos{\left (\frac{\pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}$$
           /   _________________                                         _________________                                   \      _________________                                         _________________                                   
           |14/   2        2        /2*pi\    /atan2(im(f), re(f))\   14/   2        2        /atan2(im(f), re(f))\    /2*pi\|   14/   2        2        /atan2(im(f), re(f))\    /2*pi\   14/   2        2        /atan2(im(f), re(f))\    /2*pi\
    y4 = I*|\/  im (f) + re (f) *cos|----|*sin|-------------------| - \/  im (f) + re (f) *cos|-------------------|*sin|----|| + \/  im (f) + re (f) *cos|-------------------|*cos|----| + \/  im (f) + re (f) *sin|-------------------|*sin|----|
           \                        \ 7  /    \         7         /                           \         7         /    \ 7  //                           \         7         /    \ 7  /                           \         7         /    \ 7  /
    $$y_{4} = i \left(\sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} \cos{\left (\frac{2 \pi}{7} \right )} - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{2 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}\right) + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{2 \pi}{7} \right )} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \cos{\left (\frac{2 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}$$
           /   _________________                                         _________________                                   \      _________________                                         _________________                                   
           |14/   2        2        /atan2(im(f), re(f))\    /2*pi\   14/   2        2        /2*pi\    /atan2(im(f), re(f))\|   14/   2        2        /atan2(im(f), re(f))\    /2*pi\   14/   2        2        /atan2(im(f), re(f))\    /2*pi\
    y5 = I*|\/  im (f) + re (f) *cos|-------------------|*sin|----| + \/  im (f) + re (f) *cos|----|*sin|-------------------|| + \/  im (f) + re (f) *cos|-------------------|*cos|----| - \/  im (f) + re (f) *sin|-------------------|*sin|----|
           \                        \         7         /    \ 7  /                           \ 7  /    \         7         //                           \         7         /    \ 7  /                           \         7         /    \ 7  /
    $$y_{5} = i \left(\sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} \cos{\left (\frac{2 \pi}{7} \right )} + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{2 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}\right) - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{2 \pi}{7} \right )} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \cos{\left (\frac{2 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}$$
           /     _________________                                         _________________                                   \      _________________                                         _________________                                   
           |  14/   2        2        /atan2(im(f), re(f))\    /3*pi\   14/   2        2        /3*pi\    /atan2(im(f), re(f))\|   14/   2        2        /atan2(im(f), re(f))\    /3*pi\   14/   2        2        /atan2(im(f), re(f))\    /3*pi\
    y6 = I*|- \/  im (f) + re (f) *cos|-------------------|*sin|----| - \/  im (f) + re (f) *cos|----|*sin|-------------------|| + \/  im (f) + re (f) *sin|-------------------|*sin|----| - \/  im (f) + re (f) *cos|-------------------|*cos|----|
           \                          \         7         /    \ 7  /                           \ 7  /    \         7         //                           \         7         /    \ 7  /                           \         7         /    \ 7  /
    $$y_{6} = i \left(- \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} \cos{\left (\frac{3 \pi}{7} \right )} - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{3 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}\right) + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{3 \pi}{7} \right )} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \cos{\left (\frac{3 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}$$
           /   _________________                                         _________________                                   \      _________________                                         _________________                                   
           |14/   2        2        /atan2(im(f), re(f))\    /3*pi\   14/   2        2        /3*pi\    /atan2(im(f), re(f))\|   14/   2        2        /atan2(im(f), re(f))\    /3*pi\   14/   2        2        /atan2(im(f), re(f))\    /3*pi\
    y7 = I*|\/  im (f) + re (f) *cos|-------------------|*sin|----| - \/  im (f) + re (f) *cos|----|*sin|-------------------|| - \/  im (f) + re (f) *cos|-------------------|*cos|----| - \/  im (f) + re (f) *sin|-------------------|*sin|----|
           \                        \         7         /    \ 7  /                           \ 7  /    \         7         //                           \         7         /    \ 7  /                           \         7         /    \ 7  /
    $$y_{7} = i \left(- \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} \cos{\left (\frac{3 \pi}{7} \right )} + \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{3 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}\right) - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \sin{\left (\frac{3 \pi}{7} \right )} \sin{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )} - \sqrt[14]{\left(\Re{f}\right)^{2} + \left(\Im{f}\right)^{2}} \cos{\left (\frac{3 \pi}{7} \right )} \cos{\left (\frac{1}{7} \operatorname{atan_{2}}{\left (\Im{f},\Re{f} \right )} \right )}$$
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