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Find all complex numbers z such that z^4 = -4

 May 1, 2020
 #1
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If a number is written in  r·cis( theta )  form, then its four fourth roots are:

     r1/4·cis( theta/4 ),     r1/4·cis( theta/4 + pi/4 ),     r1/4·cis( theta/4 + 2pi/4 ),     r1/4·cis( theta/4 + 3pi/4 ).

 

Writing  -4  into  r·cis( theta )  form:  -4  =  4·cis( pi )

 

So its four fourth roots are: 

      41/4·cis( pi/4 ),     41/4·cis( pi/4 + pi/4 ),     41/4·cis( pi/4 + 2pi/4 ),     41/4·cis( pi/4 + 3pi/4 ).

 

[You should simplify these answers.]

 May 1, 2020

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