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If z is nonreal root of [-1]^[1/7] then,...

If z is nonreal root of `[-1]^[1/7]` then,find the value of `z^86`+`z^175`+`z^289`

Text Solution

Verified by Experts

The correct Answer is:
`-1`
Let `z = ""^(7)sqrt(-1)`. Then
`Z^(7) = -1`
`Therefore z^(86) + z^(175) + z^(289)`
`= (z^(7))^(12) xx z^(2) + (z^(7))^(25) + (z^(7))^(41) xx z^(2)`
`= (-1)^(12) xx z^(2) + (-1)^(25) + (-1)^(41) xx z^(2)`
`= z ^(2) - 1-z^(2) = - 1`
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