Chapter 2.9, Problem 26P

Express the following complex numbers in the $x+iy$ form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and a computer to check your answers.

${\left(\frac{2i}{i+\sqrt{3}}\right)}^{19}$