How do I get the answer of second picture using the first pic's info.

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sciencesolve eNotes educator| Certified Educator

You need to remember that `z = a + b*i` and bar `z = a - b*i` , hence `z + bar z =a + b*i + a- b*i = 2a`

Since `z + bar z = 2sqrt 2` , then `2a = 2 sqrt 2`

`a = sqrt 2`

The problem provides the equation `z*bar z = 3` , such that:

`z*bar z = (a + b*i)(a - b*i) = a^2 + b^2` , since `i^2 = -1`

`a^2 + b^2 = 3`

But `a = sqrt 2,` hence `a^2 = 2`

`2 + b^2 = 3 => b^2 = 1 => b = +-1`

`z^2 = (a + b*i)^2 => z^2 = a^2 + 2abi- b^2`

`z^2 =1 + 2sqrt2*i`

`z^4 = (1 + 2sqrt2*i)^2 =1 + 4sqrt2 - 8 = -7 + 4sqrt 2`

`z^5 = (-7 + 4sqrt 2)(sqrt2 + i)`

`z^5 = -7sqrt2 - 7i + 8 + 4sqrt2*i`

`z^5 = (8 - 7sqrt 2) + (4sqrt2 - 7)*i`

`(bar z)^2 = (sqrt2 - i)^2 = 2 - 2sqrt2*i- 1`

`(bar z)^2 = 1 - 2sqrt2 *i`

`(bar z)^4 = (1- 2sqrt2*i)^2 = 1- 4sqrt2 - 8 = -7- 4sqrt 2`

`(bar z)^5 = -(7+ 4sqrt 2)(sqrt2 - i) = -7sqrt2 + 7i - 8 + 4sqrt2*i`

`(bar z)^5 = -(7sqrt2+8) 8 + (7 + 4sqrt2)*i`

Evaluate `z^5 + (bar z)^5` , such that:

`z^5 + (bar z)^5 = 8 - 7sqrt 2 + (4sqrt2 - 7)*i - 7sqrt2 - 8 + (7 + 4sqrt2)*i`

`z^5 + (bar z)^5 = -14sqrt2 + 8sqrt2*i`

Hence, evaluating `z^5 + (bar z)^5` , based on given information, yields `z^5 + (bar z)^5 = -14sqrt2 + 8sqrt2*i` .