Square root of a complex number.

Date: 6/6/96 at 12:32:47
From: Anonymous
Subject: The square root of a complex number.

My question is: Find the square root of 5+12i.  Hint: Let the square 
root of 5+12i=a+bi.

This is how the question is in my math textbook.  I have been trying 
it for a while now, but have been unable to get the answer.  I have 
tried changing it to polar form and trying to get an answer from that, 
but it does not seem to lead anywhere.

Any help you could give will be greatly appreciated.
Thanks,
Lindsey

Date: 6/14/96 at 1:2:55
From: Doctor Beth
Subject: Re: The square root of a complex number.

If the square root of a+bi is 5+12i, then you know that 
(a+bi)^2 = 5+12i.

This leads to the equation a^2 + 2abi - b^2 = 5+12i.  Whenever two 
complex numbers are equal, their real parts and imaginary parts are 
equal, so you have the 2 equations a^2 - b^2 = 5 and 2ab = 12.  Since 
this is two equations with two unknowns, you can probably take it from 
here.  (I did the problem and ended up with something that factored 
very nicely.)

-Doctor Beth,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/