Math Help!?

Question

Please help with my pre-cal homework, here is the question.
( it has symbols in it, so i have uploaded a picture of the question on my webserver)
http://www.nnrkarate.com/mathhelp.png

Answer 1

Z=x+iy, Z*=x-iy 'star' represents complex conjugate
So LHS=(Z+Z*)*= (x+iy+x-iy)* =((x+x)+i(y-y))*
using the relation (X+iY)*= X-iY above
LHS= (x+x)-i(y-y)
RHS= Z+Z*= x+iy+x-iy= (x+x)-i(y-y)
Are you sure the sum wasn't 'Prove (Z1*+Z2)*= Z1+Z2* ???
In any case the subtraction one uses the same idea!

Answer 2

troothskr has the right idea.

Let me point out that for additon is communtative and so for a) you can freely commute the z and its conjugate.

It's worth noting that
for Z = a + bi
a )Z+Z* = 2a
b) Z-Z* = 2bi

note that a) is real and so it's equal to its conjugate

for b)
(2bi)* = -2bi

Complex numbers are a lot of fun. Wait until you look a the n roots of 1. Yep, the equation

Z^n = 1 has n complex roots, known as the n roots of 1. They are equalangular on the unit circle. Kool, huh?