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Given z+(1/z)=1, what is z^10-z^(-10)?

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greenbel | Honors

Posted September 10, 2013 at 3:55 PM via web

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Given z+(1/z)=1, what is z^10-z^(-10)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted September 10, 2013 at 4:17 PM (Answer #1)

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The problem provides the information that `z + 1/z = 1` , hence, you may re-write the condition, such that:

`z^2 + 1 = z => z^2 - z + 1 = 0`

Multiplying by `z + 1` both sides, yields:

`(z + 1)(z^2 - z + 1) = 0 => z^3 + 1 = 0 => z^3 = -1`

You need to evaluate `z^10 + z^(-10)` , hence, you may use the information that `z^3 = -1` , such that:

`z^10 + z^(-10) = z^(3+3+3+1) + z^(-(3+3+3+1))`

`z^10 + z^(-10) = z^3*z^3*z^3*z + z^(-3)*z^(-3)*z^(-3)*z^(-1)`

Replacing -`1` for `z^3` yields:

`z^10 + z^(-10) = (-1)*(-1)*(-1)*z + 1/((-1)*(-1)*(-1)*z)`

`z^10 + z^(-10) = -z - 1/z`

Factoring out -1 yields:

`z^10 + z^(-10) = -(z + 1/z)`

Since the problem provides the information that `z + 1/z = 1` , yields:

`z^10 + z^(-10) = -1`

Hence, evaluating the requested expression, under the given conditions, yields `z^10 + z^(-10) = -1.`

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