What are the fourth roots of 2√3−2i ? Enter the roots in order of increasing angle measure.

Transform a + b·i  into r·cis(theta) form.     

 

r  =  sqrt( a² + b² )  =  sqrt[ ( 2sqrt(3) )² + (-2)² ]  =  sqrt[ 12 + 4 ]  =  4

theta  =  tan^-1( b/a )  =  tan^-1( -2 / ( 2sqrt(3) ) )  =  -30⁰  =  330^o

 

4 cis( 330^o )

 

To find the fourth root find the fourth root of the constant and divide the angle by 4.

 4^1/4 cis( 330^o / 4 )  =  sqrt(2)·cis( 82.5^o )

 

To find the other three roots, add 90^o to the preceding root:  sqrt(2)·cis( 172.5^o ), ...