What is the fourth root of the following complex number?

Question

Part a: find the modulus and angles of the roots of #z=1+sqrt3 i#
Part b: What are the fourth roots of z?

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Answer 1 · 2018-05-29T06:19:26

(a) #z=1+sqrt3i#

#=>z=1/2(2+2sqrt3i)#

#=>z=1/2((sqrt3)^2+i^2+2sqrt3i)#

#=>z=1/2(sqrt3+i)^2#

#=>z^(1/2)=pm1/sqrt2(sqrt3+i)#

(b)
When

#=>z^(1/2)=1/sqrt2(sqrt3+i)#

#=>z^(1/2)=1/(4sqrt2)(4sqrt3+2*2*i)#

#=>z^(1/2)=1/(4sqrt2)((sqrt3+1)^2+(i(sqrt3-1))^2+2*(sqrt3+1)*i(sqrt3-1))#

#=>z^(1/2)=1/(4sqrt2)((sqrt3+1)+(i(sqrt3-1)))^2#

#=>z^(1/4)=pm1/2^(5/4)((sqrt3+1)+(i(sqrt3-1)))#

Similarly when

#z^(1/2)=-1/sqrt2(sqrt3+i)#

#=>z^(1/4)=pmi1/2^(5/4)((sqrt3+1)+(i(sqrt3-1)))#

#=>z^(1/4)=pm1/2^(5/4)(-(sqrt3-1)+(i(sqrt3+1)))#