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| De Moivre's Theorem |
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| If 'n' be any rational number, positive or negative, then |
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| Part I: When 'n' is any integer, positive or negative then |
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| Part II: When 'n' is a fraction. |
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| Part I: |
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| Case 1: |
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| When n is a positive integer. |
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| By actual multiplication, we have |
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| Similarly by the method of induction, |
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| Case 2: |
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| When 'n' is a negative integer. |
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| Let n = -m and m > 0 |
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| Part II: |
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| Case 3: |
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| When 'n' is a fraction. |
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| where 'p' is any positive integer and 'q' is any integer. |
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| From Part I, |
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| Now taking qth root of both sides, |
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| Hence the theorem. |
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