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

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Guest May 14, 2020

#1**0 **

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

r = sqrt( a^{2} + b^{2} ) = sqrt[ ( 2sqrt(3) )^{2} + (-2)^{2} ] = sqrt[ 12 + 4 ] = 4

theta = tan^{-1}( b/a ) = tan^{-1}( -2 / ( 2sqrt(3) ) ) = -30^{0} = 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} ), ...

geno3141 May 14, 2020