SOLUTION: Simplify (i+1)^3200-(i-1)^3200

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Question 1050430: Simplify (i+1)^3200-(i-1)^3200
Answer by ikleyn(52872) About Me  (Show Source):
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Simplify (i+1)^3200-(i-1)^3200
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i + 1 = sqrt%282%29%2A%28%28sqrt%282%29%2F2%29+%2B+i%2A%28sqrt%282%29%2F2%29%29 = sqrt%282%29%2A%28cos%28pi%2F4%29+%2B+i%2Asin%28pi%2F4%29%29.

Now apply de Moivre's formula:  %28r%2A%28cos%28alpha%29+%2B+i%2Asin%28alpha%29%29%29%5En = r%5En%2A%28cos%28n%2Aalpha%29+%2B+i%2Asin%28n%2Aalpha%29%29.

You will get

%28i%2B1%29%5E3200 =  = 2%5E1600%2A%28cos%28800%2Api%29+%2B+i%2Asin%28800%2Api%29%29 = 2%5E1600.    (1)



Similarly, 

i - 1 = sqrt%282%29%2A%28%28sqrt%282%29%2F2%29+-+i%2A%28sqrt%282%29%2F2%29%29 = sqrt%282%29%2A%28cos%28pi%2F4%29+-+i%2Asin%28pi%2F4%29%29.

%28i-1%29%5E3200 =  = 2%5E1600%2A%28cos%28800%2Api%29+-+i%2Asin%28800%2Api%29%29 = 2%5E1600.    (2)



Combining (1) and (2), you will get

(i+1)^3200-(i-1)^3200 = 2%5E1600 - 2%5E1600 = 0.

Answer.  (i+1)^3200-(i-1)^3200 = 0.

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    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
    - How to take a root of a complex number
    - Solution of the quadratic equation with real coefficients on complex domain
    - How to take a square root of a complex number
    - Solution of the quadratic equation with complex coefficients on complex domain
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