Complex number in rectangular form What is (1+2j) + (1+3j)? Your answer should contain three significant figures.

Multiplicative Inverse of a Complex Number The multiplicative inverse of a complex number z is a complex number zm such that ${\displaystyle z\times zm=1}$. Find the multiplicative inverse of complex number.
${\displaystyle z=1+i}$

How can we prove that ${\displaystyle \left|z\right|\le Re\left(z\right)?}$

Calculate $$(7+9i)-(\frac{5}{7}+\frac{5i}{3})$$.

Find the product ${\displaystyle z_1{z}_2}$ and the quotient ${\displaystyle \frac{z_1}{z_2}}$. Express your answers in polar form.
${\displaystyle z_1=\frac{7}{8}(\cos \left({35}^{\circ }\right)+i\sin \left({35}^{\circ }\right)),z_2=\frac{1}{8}(\cos \left({135}^{\circ }\right)+i\sin \left({135}^{\circ }\right))}$
${\displaystyle z_1{z}_2=?}$
${\displaystyle \frac{z_1}{z_2}=?}$

Given ${\displaystyle f\left(z\right)=\frac{1}{z^2(1-z)}}$ I am to find two Laurent series expansions. There are two singularities, z=0 and z=1. So for the first expansion, I used the region 0<|z|<1 and I got ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{z}^n+\frac{1}{z}+\frac{1}{z^2}}$. The second expansion is for the region ${\displaystyle 1<\left|z\right|<\mathrm{\infty }}$. I don't know how to approach this, the explanation in my book is confusing. Any help?

Find zw and ${\displaystyle \frac{z}{w}}$. Write each answer in polar form and in exponential form.
${\displaystyle z=7(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})}$
${\displaystyle w=7(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10})}$
The product zw in polar form is ? and in exponential form is ?.

How to express ${\displaystyle e^{2-i}}$ in the form ${\displaystyle a+ib}$?