the impedance of a circuit can be calculated by the expression
Z = Z1 X Z2 / Z1 + Z2
If Z1 = 4 + j10 & Z2 = 12 - j3
how do I calculate the impedance Z in rectangular and polar forms?
First, we need to know what the formula really is. Is it
$\displaystyle Z= (Z1)\left(\frac{Z2}{Z1}\right)+ Z2$
which is what you wrote, or is it
$\displaystyle Z= \frac{Z1Z2}{Z_1+ Z_2}$?
With Z1= 4+ j10, Z2= 12- j3, Z1+ Z2= 16+ j7. Now, write the three numbers, 4+j10, 12- j3, and 16+ j7 in polar form: for 4+ j10, $\displaystyle r= \sqrt{4^2+ 10^2}= \sqrt{116}$ and [itex]\theta= arctan(10/4)= arctan(5/2)= 1.19.
Multiplying $\displaystyle r_1 cis(\theta_1)= r_1e^{i\theta_1}$ and $\displaystyle r_2 cis(\theta_2)= r_2e^{i\theta_2}$ gives $\displaystyle (r_1r_2) cis(\theta_1+ \theta_2)= (r_1r_2)e^{i(\theta_1+ \theta_2)}$