Square root of complex numbers?

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

1 Answer
Feb 5, 2018

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

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

Explanation:

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)