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Решение любых уравнений, Решение уравнений Работа проверена: Ruslana999 Время решения: 4 мин Сложность: 3.7

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Дано

$$6^{frac{25 x}{2} + 2} = frac{1}{216}$$
Подробное решение
Дано уравнение:
$$6^{frac{25 x}{2} + 2} = frac{1}{216}$$
или
$$6^{frac{25 x}{2} + 2} – frac{1}{216} = 0$$
или
$$36 left(2176782336 sqrt{6}right)^{x} = frac{1}{216}$$
или
$$left(2176782336 sqrt{6}right)^{x} = frac{1}{7776}$$
– это простейшее показательное ур-ние
Сделаем замену
$$v = left(2176782336 sqrt{6}right)^{x}$$
получим
$$v – frac{1}{7776} = 0$$
или
$$v – frac{1}{7776} = 0$$
Переносим свободные слагаемые (без v)
из левой части в правую, получим:
$$v = frac{1}{7776}$$
Получим ответ: v = 1/7776
делаем обратную замену
$$left(2176782336 sqrt{6}right)^{x} = v$$
или
$$x = frac{log{left (v right )}}{log{left (2176782336 sqrt{6} right )}}$$
Тогда, окончательный ответ
$$x_{1} = frac{log{left (frac{1}{7776} right )}}{log{left (2176782336 sqrt{6} right )}} = – frac{2}{5}$$
Ответ

/ 4/5
|6 |
2*log|—-|
6 /
x1 = ———–
log(6)

$$x_{1} = frac{2 log{left (frac{6^{frac{4}{5}}}{6} right )}}{log{left (6 right )}}$$

/ / /pi
/ _______________________________ | |sin|–|||
| / 3/5 2/pi 3/5 2/pi | | | 25/||
| / 6 *cos |–| 6 *sin |–| | 2*I*|-pi + atan|——-||
| / 25/ 25/ | | | /pi||
2*log| / ————- + ————- | | |cos|–|||
/ 6 6 / 25///
x2 = ——————————————– + ————————-
log(6) log(6)

$$x_{2} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- pi + {atan}{left (frac{sin{left (frac{pi}{25} right )}}{cos{left (frac{pi}{25} right )}} right )}right)$$

/ / /pi
/ _______________________________ | |sin|–|||
| / 3/5 2/pi 3/5 2/pi | | | 25/||
| / 6 *cos |–| 6 *sin |–| | 2*I*|pi – atan|——-||
| / 25/ 25/ | | | /pi||
2*log| / ————- + ————- | | |cos|–|||
/ 6 6 / 25///
x3 = ——————————————– + ————————
log(6) log(6)

$$x_{3} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- {atan}{left (frac{sin{left (frac{pi}{25} right )}}{cos{left (frac{pi}{25} right )}} right )} + piright)$$

/ /2*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/2*pi 3/5 2/2*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /2*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x4 = ———————————————— – ——————-
log(6) log(6)

$$x_{4} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{2 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{2 pi}{25} right )}} right )} – frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{2 pi}{25} right )}}{cos{left (frac{2 pi}{25} right )}} right )}$$

/ /2*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/2*pi 3/5 2/2*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /2*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x5 = ———————————————— + ——————-
log(6) log(6)

$$x_{5} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{2 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{2 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{2 pi}{25} right )}}{cos{left (frac{2 pi}{25} right )}} right )}$$

/ / /3*pi
/ ___________________________________ | |sin|—-|||
| / 3/5 2/3*pi 3/5 2/3*pi | | | 25 /||
| / 6 *cos |—-| 6 *sin |—-| | 2*I*|-pi + atan|———||
| / 25 / 25 / | | | /3*pi||
2*log| / ————— + ————— | | |cos|—-|||
/ 6 6 / 25 ///
x6 = ———————————————— + —————————
log(6) log(6)

$$x_{6} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{3 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{3 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- pi + {atan}{left (frac{sin{left (frac{3 pi}{25} right )}}{cos{left (frac{3 pi}{25} right )}} right )}right)$$

/ / /3*pi
/ ___________________________________ | |sin|—-|||
| / 3/5 2/3*pi 3/5 2/3*pi | | | 25 /||
| / 6 *cos |—-| 6 *sin |—-| | 2*I*|pi – atan|———||
| / 25 / 25 / | | | /3*pi||
2*log| / ————— + ————— | | |cos|—-|||
/ 6 6 / 25 ///
x7 = ———————————————— + ————————–
log(6) log(6)

$$x_{7} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{3 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{3 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- {atan}{left (frac{sin{left (frac{3 pi}{25} right )}}{cos{left (frac{3 pi}{25} right )}} right )} + piright)$$

/ /4*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/4*pi 3/5 2/4*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /4*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x8 = ———————————————— – ——————-
log(6) log(6)

$$x_{8} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{4 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{4 pi}{25} right )}} right )} – frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{4 pi}{25} right )}}{cos{left (frac{4 pi}{25} right )}} right )}$$

/ /4*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/4*pi 3/5 2/4*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /4*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x9 = ———————————————— + ——————-
log(6) log(6)

$$x_{9} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{4 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{4 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{4 pi}{25} right )}}{cos{left (frac{4 pi}{25} right )}} right )}$$

/ /6*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/6*pi 3/5 2/6*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /6*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x10 = ———————————————— – ——————-
log(6) log(6)

$$x_{10} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{6 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{6 pi}{25} right )}} right )} – frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{6 pi}{25} right )}}{cos{left (frac{6 pi}{25} right )}} right )}$$

/ /6*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/6*pi 3/5 2/6*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /6*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x11 = ———————————————— + ——————-
log(6) log(6)

$$x_{11} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{6 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{6 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{6 pi}{25} right )}}{cos{left (frac{6 pi}{25} right )}} right )}$$

/ / /7*pi
/ ___________________________________ | |sin|—-|||
| / 3/5 2/7*pi 3/5 2/7*pi | | | 25 /||
| / 6 *cos |—-| 6 *sin |—-| | 2*I*|-pi + atan|———||
| / 25 / 25 / | | | /7*pi||
2*log| / ————— + ————— | | |cos|—-|||
/ 6 6 / 25 ///
x12 = ———————————————— + —————————
log(6) log(6)

$$x_{12} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{7 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{7 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- pi + {atan}{left (frac{sin{left (frac{7 pi}{25} right )}}{cos{left (frac{7 pi}{25} right )}} right )}right)$$

/ / /7*pi
/ ___________________________________ | |sin|—-|||
| / 3/5 2/7*pi 3/5 2/7*pi | | | 25 /||
| / 6 *cos |—-| 6 *sin |—-| | 2*I*|pi – atan|———||
| / 25 / 25 / | | | /7*pi||
2*log| / ————— + ————— | | |cos|—-|||
/ 6 6 / 25 ///
x13 = ———————————————— + ————————–
log(6) log(6)

$$x_{13} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{7 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{7 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- {atan}{left (frac{sin{left (frac{7 pi}{25} right )}}{cos{left (frac{7 pi}{25} right )}} right )} + piright)$$

/ /8*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/8*pi 3/5 2/8*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /8*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x14 = ———————————————— – ——————-
log(6) log(6)

$$x_{14} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{8 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{8 pi}{25} right )}} right )} – frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{8 pi}{25} right )}}{cos{left (frac{8 pi}{25} right )}} right )}$$

/ /8*pi
/ ___________________________________ |sin|—-||
| / 3/5 2/8*pi 3/5 2/8*pi | | 25 /|
| / 6 *cos |—-| 6 *sin |—-| | 2*I*atan|———|
| / 25 / 25 / | | /8*pi|
2*log| / ————— + ————— | |cos|—-||
/ 6 6 / 25 //
x15 = ———————————————— + ——————-
log(6) log(6)

$$x_{15} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{8 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{8 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{8 pi}{25} right )}}{cos{left (frac{8 pi}{25} right )}} right )}$$

/ / /9*pi
/ ___________________________________ | |sin|—-|||
| / 3/5 2/9*pi 3/5 2/9*pi | | | 25 /||
| / 6 *cos |—-| 6 *sin |—-| | 2*I*|-pi + atan|———||
| / 25 / 25 / | | | /9*pi||
2*log| / ————— + ————— | | |cos|—-|||
/ 6 6 / 25 ///
x16 = ———————————————— + —————————
log(6) log(6)

$$x_{16} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{9 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{9 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- pi + {atan}{left (frac{sin{left (frac{9 pi}{25} right )}}{cos{left (frac{9 pi}{25} right )}} right )}right)$$

/ / /9*pi
/ ___________________________________ | |sin|—-|||
| / 3/5 2/9*pi 3/5 2/9*pi | | | 25 /||
| / 6 *cos |—-| 6 *sin |—-| | 2*I*|pi – atan|———||
| / 25 / 25 / | | | /9*pi||
2*log| / ————— + ————— | | |cos|—-|||
/ 6 6 / 25 ///
x17 = ———————————————— + ————————–
log(6) log(6)

$$x_{17} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{9 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{9 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- {atan}{left (frac{sin{left (frac{9 pi}{25} right )}}{cos{left (frac{9 pi}{25} right )}} right )} + piright)$$

/ / /11*pi
/ _____________________________________ | |sin|—–|||
| / 3/5 2/11*pi 3/5 2/11*pi | | | 25 /||
| / 6 *cos |—–| 6 *sin |—–| | 2*I*|-pi + atan|———-||
| / 25 / 25 / | | | /11*pi||
2*log| / —————- + —————- | | |cos|—–|||
/ 6 6 / 25 ///
x18 = ————————————————– + —————————-
log(6) log(6)

$$x_{18} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{11 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{11 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- pi + {atan}{left (frac{sin{left (frac{11 pi}{25} right )}}{cos{left (frac{11 pi}{25} right )}} right )}right)$$

/ / /11*pi
/ _____________________________________ | |sin|—–|||
| / 3/5 2/11*pi 3/5 2/11*pi | | | 25 /||
| / 6 *cos |—–| 6 *sin |—–| | 2*I*|pi – atan|———-||
| / 25 / 25 / | | | /11*pi||
2*log| / —————- + —————- | | |cos|—–|||
/ 6 6 / 25 ///
x19 = ————————————————– + —————————
log(6) log(6)

$$x_{19} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{11 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{11 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} left(- {atan}{left (frac{sin{left (frac{11 pi}{25} right )}}{cos{left (frac{11 pi}{25} right )}} right )} + piright)$$

/ /12*pi
/ _____________________________________ |sin|—–||
| / 3/5 2/12*pi 3/5 2/12*pi | | 25 /|
| / 6 *cos |—–| 6 *sin |—–| | 2*I*atan|———-|
| / 25 / 25 / | | /12*pi|
2*log| / —————- + —————- | |cos|—–||
/ 6 6 / 25 //
x20 = ————————————————– – ——————–
log(6) log(6)

$$x_{20} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{12 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{12 pi}{25} right )}} right )} – frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{12 pi}{25} right )}}{cos{left (frac{12 pi}{25} right )}} right )}$$

/ /12*pi
/ _____________________________________ |sin|—–||
| / 3/5 2/12*pi 3/5 2/12*pi | | 25 /|
| / 6 *cos |—–| 6 *sin |—–| | 2*I*atan|———-|
| / 25 / 25 / | | /12*pi|
2*log| / —————- + —————- | |cos|—–||
/ 6 6 / 25 //
x21 = ————————————————– + ——————–
log(6) log(6)

$$x_{21} = frac{2}{log{left (6 right )}} log{left (sqrt{frac{6^{frac{3}{5}}}{6} cos^{2}{left (frac{12 pi}{25} right )} + frac{6^{frac{3}{5}}}{6} sin^{2}{left (frac{12 pi}{25} right )}} right )} + frac{2 i}{log{left (6 right )}} {atan}{left (frac{sin{left (frac{12 pi}{25} right )}}{cos{left (frac{12 pi}{25} right )}} right )}$$

/ ___________
| 3/10 4/5 / ___ |
/ ___ 3/10 |2*2 *3 */ 5 + / 5 |
|/ 6 *6 | 2*I*atan|—————————|
2*log|———–| | 4/5 ___ 4/5 |
6 / – 6 + / 5 *6 /
x22 = —————— – ————————————-
log(6) log(6)

$$x_{22} = frac{2 log{left (frac{6^{frac{4}{5}}}{6} right )}}{log{left (6 right )}} – frac{2 i}{log{left (6 right )}} {atan}{left (frac{2 cdot 2^{frac{3}{10}} cdot 3^{frac{4}{5}} sqrt{sqrt{5} + 5}}{- 6^{frac{4}{5}} + sqrt{5} cdot 6^{frac{4}{5}}} right )}$$

/ ___________
| 3/10 4/5 / ___ |
/ ___ 3/10 |2*2 *3 */ 5 + / 5 |
|/ 6 *6 | 2*I*atan|—————————|
2*log|———–| | 4/5 ___ 4/5 |
6 / – 6 + / 5 *6 /
x23 = —————— + ————————————-
log(6) log(6)

$$x_{23} = frac{2 log{left (frac{6^{frac{4}{5}}}{6} right )}}{log{left (6 right )}} + frac{2 i}{log{left (6 right )}} {atan}{left (frac{2 cdot 2^{frac{3}{10}} cdot 3^{frac{4}{5}} sqrt{sqrt{5} + 5}}{- 6^{frac{4}{5}} + sqrt{5} cdot 6^{frac{4}{5}}} right )}$$

/ / ___________
| | 3/10 4/5 / ___ ||
/ ___ 3/10 | |2*2 *3 */ 5 – / 5 ||
|/ 6 *6 | 2*I*|-pi – atan|—————————||
2*log|———–| | | 4/5 ___ 4/5 ||
6 / – 6 – / 5 *6 //
x24 = —————— + ———————————————
log(6) log(6)

$$x_{24} = frac{2 log{left (frac{6^{frac{4}{5}}}{6} right )}}{log{left (6 right )}} + frac{2 i}{log{left (6 right )}} left(- pi – {atan}{left (frac{2 cdot 2^{frac{3}{10}} cdot 3^{frac{4}{5}} sqrt{- sqrt{5} + 5}}{- sqrt{5} cdot 6^{frac{4}{5}} – 6^{frac{4}{5}}} right )}right)$$

/ / ___________
| | 3/10 4/5 / ___ ||
/ ___ 3/10 | |2*2 *3 */ 5 – / 5 ||
|/ 6 *6 | 2*I*|pi + atan|—————————||
2*log|———–| | | 4/5 ___ 4/5 ||
6 / – 6 – / 5 *6 //
x25 = —————— + ——————————————–
log(6) log(6)

$$x_{25} = frac{2 log{left (frac{6^{frac{4}{5}}}{6} right )}}{log{left (6 right )}} + frac{2 i}{log{left (6 right )}} left({atan}{left (frac{2 cdot 2^{frac{3}{10}} cdot 3^{frac{4}{5}} sqrt{- sqrt{5} + 5}}{- sqrt{5} cdot 6^{frac{4}{5}} – 6^{frac{4}{5}}} right )} + piright)$$
Численный ответ

x1 = -0.400000000000000

x2 = -0.4 – 3.36644398898649*i

x3 = -0.4 + 3.36644398898649*i

x4 = -0.4 – 0.280536999082207*i

x5 = -0.4 + 0.280536999082207*i

x6 = -0.4 – 3.08590698990428*i

x7 = -0.4 + 3.08590698990428*i

x8 = -0.4 – 0.561073998164414*i

x9 = -0.4 + 0.561073998164414*i

x10 = -0.4 – 0.841610997246622*i

x11 = -0.4 + 0.841610997246622*i

x12 = -0.4 – 2.52483299173986*i

x13 = -0.4 + 2.52483299173986*i

x14 = -0.4 – 1.12214799632883*i

x15 = -0.4 + 1.12214799632883*i

x16 = -0.4 – 2.24429599265766*i

x17 = -0.4 + 2.24429599265766*i

x18 = -0.4 – 1.96375899357545*i

x19 = -0.4 + 1.96375899357545*i

x20 = -0.4 – 1.68322199449324*i

x21 = -0.4 + 1.68322199449324*i

x22 = -0.4 – 1.40268499541104*i

x23 = -0.4 + 1.40268499541104*i

x24 = -0.4 – 2.80536999082207*i

x25 = -0.4 + 2.80536999082207*i

   

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