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What is product (1-i)(1-i^2)(1-i^3)______(1-i^2013)?

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ruals | Salutatorian

Posted June 27, 2013 at 3:48 PM via web

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What is product (1-i)(1-i^2)(1-i^3)______(1-i^2013)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 27, 2013 at 4:06 PM (Answer #1)

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You need to use the powers of imaginary number `i` , such that:

`{(i = i),(i^2 = -1),(i^3 = -i),(i^4 = 1):}`

Replacing the powers of `i` in the given product yields:

`(1 - i)(1 - i^2)(1 - i^3)(1 - i^4)....(1 - i^2013) = (1 - i)(1 + 1)(1 + i)(1 - 1)....(1 - i^2013)`

Since the factor `(1 - 1) = 0` , hence, using the zero product rule, then  evaluating the product yields 0.

Hence, evaluating the given product, using the powers of imaginary number `i` , yields `(1 - i)(1 - i^2)(1 - i^3)(1 - i^4)....(1 - i^2013) = 0.`

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