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# What are the fourth roots of ?

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What are the fourth roots of $$-3+3\sqrt{3}i$$?

Enter your answer by filling in the boxes. Enter the roots in order of increasing angle measure in the simplest form.

___cis(___)

Jun 10, 2020

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To find the primary fourth root of the complex number  a + b·i,  place the number into  r·cis( theta ) form.

r  =  sqrt( a2 + b2 )

theta  =  tan-1( b/a )    [make certain that the angle is in the correct quadrant; in this case, the 2nd[

The primary fourth root will be:  r1/4·cis( theta/4 ).

To find the other three roots, change the angle by adding  pi/2, pi, and 3pi/2 to the original angle.

Jun 10, 2020