Square root of complex numbers?

Question

Find the square roots of complex numbers
#z=144/(5-2i)#

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Answer 1 · 2018-02-05T15:55:51

#sqrt(144/(5-2i))=12/root(4)29(cos(theta/2)+isin(theta/2))#

where #theta=tan^(-1)(2/5)#

#sqrt(144/(5-2i))=sqrt((144(5+2i))/((5-2i)(5+2i))#

= #12sqrt((5+2i)/29)=12/sqrt29sqrt(5+2i)#

Let #5+2i=r(costheta+isintheta)#,

then #r=sqrt(5^2+2^2)=sqrt29# and #theta=tan^(-1)(2/5)#

and hence usin DeMoivre's theorem

#sqrt(5+2i)=root(4)29(cos(theta/2)+isin(theta/2))#

Hence #sqrt(144/(5-2i))=12/sqrt29root(4)29(cos(theta/2)+isin(theta/2))#

= #12/root(4)29(cos(theta/2)+isin(theta/2))#

where #theta=tan^(-1)(2/5)#