Find the principal argument of the complex number `((1+i)^5(1+sqrt(3i))^2)/(-1i(-sqrt(3)+i))`

Explanation

`z = ((1+i)^(5)(1+sqrt(3i))^(2))/(-2i(-sqrt(3)+i))`
`(sqrt(2)^(5)((1)/(sqrt(2))+(i)/(sqrt(2))).2^(2)((1)/(2)+(sqrt(3))/(2)i))/((2i)2(sqrt(3)/(3)-(i)/(2)))`
`5 arg((1)/(sqrt(2))+(1)/(sqrt(2)))+2 arg((1)/(2)+(sqrt(3))/(2)i)`
`therefore arg(Z) - arg(i) - arg((sqrt(3))/(2)-(i)/(2))" "(because arg(sqrt(2)^(5)),arg(2^(2)), arg (4)=0)`
`=(5pi)/(4) + (2pi)/(3)-(pi)/(2)+(pi)/(6) = (19pi)/(12)= 2pi - (5pi)/(12)`
Therefore, Z lies in the fourth quadrant.
Hence, pricncipal argument is `- (5pi)/(12)`