Решение уравнений/ Любые/ x^8=-36

Произведение корней x^8=-36

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    Решение

    Сумма и произведение корней [src]
    сумма
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             ___________                ___________              ___________                ___________                ___________                ___________              ___________                ___________     | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2       | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2       | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2       |   3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  
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  4 ___    /  1   \/ 2       4 ___    /  1   \/ 2     4 ___    /  1   \/ 2       4 ___    /  1   \/ 2       4 ___    /  1   \/ 2       4 ___    /  1   \/ 2     4 ___    /  1   \/ 2       4 ___    /  1   \/ 2       |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4        |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4        |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4        |             \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   
- \/ 6 *  /   - - -----  + I*\/ 6 *  /   - + -----  + \/ 6 *  /   - - -----  - I*\/ 6 *  /   - + -----  + - \/ 6 *  /   - + -----  - I*\/ 6 *  /   - - -----  + \/ 6 *  /   - + -----  + I*\/ 6 *  /   - - -----  + I*|--------------------------- + ---------------------------| + --------------------------- - --------------------------- + I*|--------------------------- - ---------------------------| + --------------------------- + --------------------------- + I*|--------------------------- - ---------------------------| - --------------------------- - --------------------------- + I*|- --------------------------- - ---------------------------| + --------------------------- - ---------------------------
        \/    2     4              \/    2     4            \/    2     4              \/    2     4              \/    2     4              \/    2     4            \/    2     4              \/    2     4        \             2                             2             /                2                             2                  \             2                             2             /                2                             2                  \             2                             2             /                2                             2                  \               2                             2             /                2                             2
    (−2343424+122+2343412−242+i(−2343424+122−2343412−242))+((−2343424+122−2343412−242+i(−2343412−242+2343424+122))+((2343412−242+2343424+122+i(−2343424+122+2343412−242))+((((−6424+12−64i12−24)+((6412−24−64i24+12)+(−6412−24+64i24+12)))+(6424+12+64i12−24))+(−2343412−242+2343424+122+i(2343412−242+2343424+122)))))\left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) + \left(\left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(\left(\left(- \sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(\sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(- \sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right)\right) + \left(\sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) + \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)\right)\right)\right)
    =
      /                ___________                   ___________\     /                ___________                   ___________\     /                ___________                   ___________\     /                  ___________                   ___________\
  |               /       ___                   /       ___ |     |               /       ___                   /       ___ |     |               /       ___                   /       ___ |     |                 /       ___                   /       ___ |
  | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |     | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |     | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |     |   3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |
  |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |     |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |     |2   *\/ 3 *  /   - + -----    2   *\/ 3 *  /   - - ----- |     |  2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |
  |           \/    2     4                 \/    2     4   |     |           \/    2     4                 \/    2     4   |     |           \/    2     4                 \/    2     4   |     |             \/    2     4                 \/    2     4   |
I*|--------------------------- + ---------------------------| + I*|--------------------------- - ---------------------------| + I*|--------------------------- - ---------------------------| + I*|- --------------------------- - ---------------------------|
  \             2                             2             /     \             2                             2             /     \             2                             2             /     \               2                             2             /
    i(−2343424+122−2343412−242)+i(−2343424+122+2343412−242)+i(−2343412−242+2343424+122)+i(2343412−242+2343424+122)i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) + i \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)
    произведение
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                                                                                                                                                                                                                    |  |               /       ___                   /       ___ |                  /       ___                   /       ___ | |  |               /       ___                   /       ___ |                  /       ___                   /       ___ | |  |               /       ___                   /       ___ |                  /       ___                   /       ___ | |  |                 /       ___                   /       ___ |                  /       ___                   /       ___ |
/             ___________                ___________\ /           ___________                ___________\ /             ___________                ___________\ /           ___________                ___________\ |  | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  | |  | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  | |  | 3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  | |  |   3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |    3/4 4 ___    /  1   \/ 2      3/4 4 ___    /  1   \/ 2  |
|            /       ___                /       ___ | |          /       ___                /       ___ | |            /       ___                /       ___ | |          /       ___                /       ___ | |  |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 3 *  /   - + -----    2   *\/ 3 *  /   - - ----- | |  |2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- | |  |2   *\/ 3 *  /   - + -----    2   *\/ 3 *  /   - - ----- |   2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- | |  |  2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |   2   *\/ 3 *  /   - - -----    2   *\/ 3 *  /   - + ----- |
|  4 ___    /  1   \/ 2       4 ___    /  1   \/ 2  | |4 ___    /  1   \/ 2       4 ___    /  1   \/ 2  | |  4 ___    /  1   \/ 2       4 ___    /  1   \/ 2  | |4 ___    /  1   \/ 2       4 ___    /  1   \/ 2  | |  |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   | |  |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   | |  |           \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   | |  |             \/    2     4                 \/    2     4   |              \/    2     4                 \/    2     4   |
|- \/ 6 *  /   - - -----  + I*\/ 6 *  /   - + ----- |*|\/ 6 *  /   - - -----  - I*\/ 6 *  /   - + ----- |*|- \/ 6 *  /   - + -----  - I*\/ 6 *  /   - - ----- |*|\/ 6 *  /   - + -----  + I*\/ 6 *  /   - - ----- |*|I*|--------------------------- + ---------------------------| + --------------------------- - ---------------------------|*|I*|--------------------------- - ---------------------------| + --------------------------- + ---------------------------|*|I*|--------------------------- - ---------------------------| - --------------------------- - ---------------------------|*|I*|- --------------------------- - ---------------------------| + --------------------------- - ---------------------------|
\        \/    2     4              \/    2     4   / \      \/    2     4              \/    2     4   / \        \/    2     4              \/    2     4   / \      \/    2     4              \/    2     4   / \  \             2                             2             /                2                             2             / \  \             2                             2             /                2                             2             / \  \             2                             2             /                2                             2             / \  \               2                             2             /                2                             2             /
    (−6412−24+64i24+12)(6412−24−64i24+12)(−6424+12−64i12−24)(6424+12+64i12−24)(−2343412−242+2343424+122+i(2343412−242+2343424+122))(2343412−242+2343424+122+i(−2343424+122+2343412−242))(−2343424+122−2343412−242+i(−2343412−242+2343424+122))(−2343424+122+2343412−242+i(−2343424+122−2343412−242))\left(- \sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)
    =
    36
    3636