1040=\frac{120}{a}+\frac{120}{a^{2}}+\frac{120}{a^{2}\cdot a}+\frac{1120}{a^{2}\cdot a^{2}}

asked by guest
on Jan 14, 2025 at 2:36 pm



You asked:

Solve the equation \(1040 = \frac{120}{a} + \frac{120}{{a}^{2}} + \frac{120}{{a}^{2} a} + \frac{1120}{{a}^{2} {a}^{2}}\) for the variable \(a\).

MathBot Answer:

The complex solutions are: \[\left\{\frac{3}{104} + \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \sqrt{\left|{- 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}} + \frac{217}{5408} + \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{17187}{562432 \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}}}}\right|}, - \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{3}{104} - i \sqrt{\left|{- \frac{17187}{562432 \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}}} - 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}} + \frac{217}{5408} + \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}}}\right|}, - \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{3}{104} + i \sqrt{\left|{- \frac{17187}{562432 \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}}} - 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}} + \frac{217}{5408} + \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}}}\right|}, - \sqrt{\left|{- 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}} + \frac{217}{5408} + \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{17187}{562432 \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}}}}\right|} + \frac{3}{104} + \sqrt{- \frac{1459}{32448 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}} + \frac{217}{10816} + 2 \sqrt[3]{- \frac{5835}{17997824} + \frac{\sqrt{300909400809}}{161980416}}}\right\} \setminus \left\{0\right\}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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