Дано
$$x^{3} – y^{3} = 91$$
Ответ
$$x_{1} = i sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} sin{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )} + sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} cos{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )}$$
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| / 2 2 / / 3 2 3 2 / 2 2 / / 3 2 3 2 | / 2 2 / / 3 2 3 2 / 2 2 / / 3 2 3 2
| 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/| ___ 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/|| 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/| ___ 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/|
| / – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *sin|————————————————————–| / 3 */ – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *cos|————————————————————–|| / – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *cos|————————————————————–| / 3 */ – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *sin|————————————————————–|
| 3 / 3 /| 3 / 3 /
x2 = I*|- —————————————————————————————————————————————- – ———————————————————————————————————————————————-| – —————————————————————————————————————————————- + ———————————————————————————————————————————————-
2 2 / 2 2
$$x_{2} = i left(- frac{1}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} sin{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )} – frac{sqrt{3}}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} cos{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )}right) + frac{sqrt{3}}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} sin{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )} – frac{1}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} cos{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )}$$
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| / 2 2 / / 3 2 3 2 / 2 2 / / 3 2 3 2 | / 2 2 / / 3 2 3 2 / 2 2 / / 3 2 3 2
| 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/| ___ 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/|| 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/| ___ 6 / / 3 2 / 3 2 |atan2 – im (y) + 3*re (y)*im(y), 91 + re (y) – 3*im (y)*re(y)/|
| / – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *sin|————————————————————–| / 3 */ – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *cos|————————————————————–|| / – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *cos|————————————————————–| / 3 */ – im (y) + 3*re (y)*im(y)/ + 91 + re (y) – 3*im (y)*re(y)/ *sin|————————————————————–|
| 3 / 3 /| 3 / 3 /
x3 = I*|- —————————————————————————————————————————————- + ———————————————————————————————————————————————-| – —————————————————————————————————————————————- – ———————————————————————————————————————————————-
2 2 / 2 2
$$x_{3} = i left(- frac{1}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} sin{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )} + frac{sqrt{3}}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} cos{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )}right) – frac{sqrt{3}}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} sin{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )} – frac{1}{2} sqrt[6]{left(3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3}right)^{2} + left(left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91right)^{2}} cos{left (frac{1}{3} {atan_{2}}{left (3 left(Re{y}right)^{2} Im{y} – left(Im{y}right)^{3},left(Re{y}right)^{3} – 3 Re{y} left(Im{y}right)^{2} + 91 right )} right )}$$
