Найдите произведение корней уравнения x^8=-36 (х в степени 8 равно минус 36) [Есть ответ!]
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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- \/ 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
    $$\left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) + \left(\left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(\left(\left(- \sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(\sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(- \sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right)\right) + \left(\sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) + \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)\right)\right)\right)$$
    =
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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  |
  |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             /
    $$i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) + i \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)$$
    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  |
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\        \/    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             /
    $$\left(- \sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt[4]{6} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - \sqrt[4]{6} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt[4]{6} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt[4]{6} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(\frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{2^{\frac{3}{4}} \sqrt[4]{3} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)$$
    =
    36
    $$36$$