One way of solving this problem is to simplify the expression and then transform into polar coordinates.
It is better if we transform each of the expressions into polar coordinates and then calculate the modulus and the argument separately.
Let z1 =1+2j,
z2 = 4-3j,
z3 = 3+4j,
z4 = 2-j.
We will denote the modulus of the z1 to z4 with R1 to R4 and will denote the arguments with the corresponding indexes ∝1 to ∝4.
We will calculate R1 and ∝1 first.
We will keep these in this form, so we can finally use a calculator, like TI 30 or similar to calculate the value of the angle.
The same way we calculate the remaining values:
This is because the complex number is in quadrant 4, with positive real part and negative imaginary part.
We are ready to calculate R and ∝.
Let write z first again:
z = [z12 • z23] / [z34 • z43 ].
The modulus R will follow the formula for z:
R =[R12 • R23 ]/[R34 • R43 ]
R =[√52 • 53 ]/[54 • √5 3].
You can simplify this answer to verify the value.
The value of the argument ∝ follows different path. It is the multiplied sum of the corresponding tan-1 by the power of the complex number zi.
∝ = 2• ∝1 + 3•∝2 - 4•∝3 - 3•∝4
∝ = 2• tan-1(2/1) + 3•(-tan-1(3/4)) - 4•tan-1(4/3) - 3•(-tan-1(1/2)).
Use a calculator to verify the answers.
This problem is long and tedious. I suggest you break similar problems into peaces and make clear notes of what you are doing. Organizing this type of problems and using a consistent notation will help you trace and check your answers.