Whenever you get a new mathematical object (like complex numbers),
you need to develop tools to work with the new object.
Arithmetic with complex numbers was introduced in a prior web exercise:
Arithmetic with Complex Numbers in the Algebra II materials.
Quickly read through this earlier section
and do plenty of the online exercises to make sure that you understand all the concepts there.
For your convenience, this current section gives a quick summary of important concepts from the
earlier section, and also discusses two new topics:
the complex conjugate, and division of complex numbers.
ARITHMETIC WITH COMPLEX NUMBERS
adding, subtracting, multiplying, dividing, complex conjugate, more
Let $\,a\,$, $\,b\,$, $\,c\,$, and
$\ d\ $ be real numbers, and let
$\,i:=\sqrt{1}\,$.

Let $z = 2  3i\,$ and $\,w = 5 + 7i\,$.
Then:
$$
\begin{alignat}{2}
z + w \ &= (23i) + (5 + 7i) \ =\ (25) + (3 + 7)i \ =\ 3 + 4i\cr
z  w \ &= (23i)  (5 + 7i) \ =\ (2+5) + (3  7)i \ =\ 7  10i\cr\cr
zw\ &= (2  3i)(5 + 7i) \ =\ 10 + 14i + 15i 21i^2 \ =\ (10 + 21) + 29i \ =\ 11 + 29i\cr\cr
\frac{z}{w}\ &= \frac{23i}{5 + 7i}\cdot\frac{57i}{57i}\ =\ \frac{(23i)(57i)}{25 + 49}\ = \frac{(1021) + (1514)i}{74}\ =\ \frac{31  29i}{74}\ =\ \frac{31}{74}  \frac{29}{74}i\cr\cr\cr
\overline{z}\ &= \overline{23i} \ =\ 2 + 31\cr\cr
\overline{w}\ &= \overline{5 + 7i}\ =\ 5  7i\cr\cr
z\overline{z}\ &= (23i)(2+3i) \ =\ 4 + 6i  6i  9i^2\ =\ 4 + 9 \ =\ 13\cr\cr
4z + \overline{2w}\ &= 4(2  3i) + \overline{2(5 + 7i)}\ =\ 8 + 12i + (\overline{10 + 14i})\ =\ 8 + 12i + ( 10  14i)\ =\ 18  2i
\end{alignat}
$$
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
