Find the complex number satisfying the system of equations `z^3+ omega^7=0a n dz^5omega^(11)=1.`

Explanation

`z^(3)+bar(omega)^(7)=0`
`impliesz^(3)=-bar(omega)^(7)`
`implies|z|^(3)=|-bar(omega)|^(7)=|omega|^(7)`
`implies|z|^(15)=|omega|^(35)`
Also, `z^(5)omega^(11)=1`
`implies|z|^(5)|omega|^(11)=1`
`implies|z|^(15)|omega|^(33)=1`
From (1) and (2), we have
`|z|=|omega|=1`
Again, `bar(omega)^(7)=-z^(3)" and "omega^(11)=z^(-5)`
`impliesbar(omega)^(77).omega^(77)=-z^(33).z^(-35)`
`impliesz^(2)=-1=i^(2)impliesz=+-i`